,,the ,,philosophical and ,,mathematical ,,commentaries ,,of ,,proclus1 ,,on ,,the ,,first ,,book ,,of ,,euclid',s ,,elements4 ,,to ,,which ,,are ,,added1 ,a ,history of the ,restoration of ,,platonic ,,theology1 ,,by ,,the ,,latter ,,platonists3 ,and a ,translation from the ,greek of ,,proclus',s ,,theological ,,elements4 ,,in ,,two ,,volumes4 ,illustration ,,vol ,i4 ,,london1 ,,printed ,,for ,,the ,,author3 ,and ,sold by ,t4 ,,payne and ,,son2 ,b4 ,,white and ,,son2 ,j4 ,,robson2 ,t4 ,,cadell2 ,,leigh and ,,co42 ,g4 ,,nicol2 ,r4 ,,faulder2 and ,t4 and ,j4 ,,egerton4 #agib4 ,price ,two ,guineas in ,boards4 ,extracts from ,curiosities of ,literature4 ,second ,edition4 ,printed for ,murray4 ,page #che4 ,mr4 ,t4 ,,taylor1 the ,platonic ,philosopher and the modern ,plethon1 consonant to that philosophy1 professes ,polytheism4#a ,the ,reader is requested to correct the following ,errors4 ,page #d4 of the ,dissertation1 ,vol4 ,i4 line #h4 for admitted1 read omitted4 ,page #af4 line #h4 for from1 read form4 ,page #ea4 ,vol4 ,i4 of the ,commentaries1 line #af4 instead of ,they are surely not the1 c4 read ,for surely it cannot be said that there are1 c4 ,line #ag4 for but we1 read but that we4 ,and line #ai4 for is by much prior to1 read is by a much greater priority4 ,vol4 ,,ii4 page #ah4 line #bf4 for and one is1 read and one part is4 ,and line #bg4 for another1 read the other4 ,page #aad4 line #ac4 for ,the angle1 read ,let the angle2 and instead of is bisected1 in the same line1 read be bisected4 ,page #daa4 line #b4 for is filled with intellect1 read fills intellect4 ,and line #c4 for it also participates1 read also it participates4 ,,to ,,the ,,sacred ,,majesty ,,of ,,truth4 ,,preface4 ,the design of the present work is to bring us acquainted with the nature and end of ,mathematics in general1 and of ,geometry in particular3 and in the execution of this design our ,author has displayed an uncommon elegance of composition1 and a most valuable store of recondite learning4 ,he is not content with every where unfolding the full1 and most accurate meaning of ,euclid2 but he continually rises in his discourse1 and leads us into the depths of the ,pythagoric and ,platonic philosophy4 ,we are surprised to find an use in ,geometry1 which at present it is by no means suspected to afford4 ,for who would conceive that it is the genuine passage to true theology1 and the vestibule of divinity8 ,this1 indeed1 is by no means the case when it is studied for lucre1 and applied to mechanical purposes2 for then the soul is neither elevated nor enlightened2 but degraded and filled with material darkness4 ,hence these ,commentaries are alone valuable to the liberal part of mankind1 who look beyond sense for certainty2 and who prefer things desirable for their own sakes1 before such as minister to the necessities of life4 ,the translation of this work is attended with great difficulty and labour2 not only from the sense of the philosopher1 which is always profound1 and frequently obscure1 but from the great incorrectness of the ,greek edition1 in which1 exclusive of numberless typographical errors1 entire sentences1 essential to the connection1 are frequently omitted2 and in one place two pages of the ,latin translation are wanting in the original1 as will be shewn in our following notes4 ,indeed1 the ,latin translation of ,francis ,barocius the ,venetian1 7,patav4 #aefj47 which was made from a variety of manuscripts1 is inconceivably valuable2 for the diagrams1 so necessary to a work of this kind1 but which are omitted in the ,greek1 are here inserted2 and the version is every where faithful1 and sufficiently perspicuous to those who are conversant in the ancient philosophy4 ,barocius justly cautions the reader not to compare his version with the printed ,greek1 which he observes is rather lacerated than printed2 as indeed1 without his translation1 it is impossible for any one to read the half of this invaluable work1 even though he should be as perfect in ,greek as in his native tongue4 ,if ,i had not1 therefore1 fortunately acquired this translation1 which is at present very rare1 ,i would have by no means engaged in this arduous undertaking4 ,barocius1 indeed1 gives evident proofs of his possessing the philosophical genius1 by the excellence of his translation1 and his preface to the reader2 and it is greatly to be lamented that he did not adorn his version with explanatory notes1 which this profound work frequently demands1 and which he was doubtless well qualified to accomplish4 ,this defect ,i have endeavoured1 as far as ,i was able1 to supply2 and at the same time have been cautious neither to weary the reader by prolixity1 nor by too much brevity to leave him destitute of proper information4 ,in the distribution of the first book of this work into chapters1 ,i have followed the order of ,barocius1 because it is natural and obvious2 and must beg leave to solicit the reader's indulgence for using the words partible and impartible1 differently from their common signification4 ,these words ,i have generally employed to express the meaning of #b and #c in the ,greek1 as ,i do not conceive that the words divisible and indivisible always convey their full signification4 ,i have likewise used quadrangle instead of square1 and quinquangle for the word pentagon4 ,for if be rendered triangle1 why should not be rendered quadrangle8 ,and1 as ,barocius observes1 why1 for a similar reason1 should not and be rendered quinquangle and sexangle2 and so of the rest8 ,uniformity is always desirable when it can be obtained2 and is no where so necessary as in scientifical disquisitions4 ,it is likewise necessary to inform the reader1 that though ,i have always endeavoured to give the faithful meaning of my ,author1 yet ,i have occasionally paraphrased his sense1 when most obscure1 and added such elucidations of my own1 as ,i either thought necessary to the full comprehension of his matter2 or which were naturally excited by the fire and spirit of the ,original4 ,if it shall appear that ,i have succeeded in the execution of this work1 and rendered it intelligible to the lovers of truth1 ,i shall rejoice in my success1 and consider my labours sufficiently rewarded4 ,the applause of the multitude ,i am neither likely nor desirous to gain2 but ,i am anxious to procure the approbation of the discerning few1 who know that the age of philosophy is past2 and who esteem the works of her ancient heroes as the most precious treasures which have escaped the ravages of time4 99 99 99 99 99 ,time1 indeed1 is like a deep and rapid river2 whatever is trifling and light1 is precipitately borne on its surface1 and what is valuable and weighty1 sinks to its bottom4 ,hence1 the superficial observer collects nothing more than the rubbish1 which it is forever devolving into the abyss of oblivion2 while the profound and contemplative genius explores the depths of the stream1 and accounts himself happy if he can gather any of the pearl which its bottom contains4 ,thus the discoveries of experimental philosophy1 float like straws on the surface1 while the wisdom of ,pythagoras and ,plato lies concealed in the depths of the river4 ,i am well aware it will be said1 that the reverse of this similitude is true2 that the modern philosophy is the pearl1 and the ancient the stubble2 and that the former will be celebrated by posterity1 and increase in reputation when the latter shall scarcely be known4 ,but let us attentively examine the truth of this assertion1 and shut our ears to the unsubstantial echoes of popular applause4 ,is it reasonable to suppose that men of such exalted abilities1 as the ,pythagoric and ,platonic philosophers possessed1 even in the estimation of their opponents1 accompanied with the greatest advantages of birth and fortune1 and the most unwearied attention1 have discovered nothing valuable1 and have left nothing behind them1 but jargon and reveries8 ,is it to be supposed1 that in an age when philosophy was almost adored2 when it was esteemed by kings1 cultivated by noblemen1 and even reverenced by the vulgar2 when empire was relinquished for its pursuit1 and every danger encountered for its possession3 is it to be supposed1 that nothing but delusion was the offspring of so glorious a period1 and nothing but folly the reward of such generous endurance8 ,or shall we say1 that the discovery of truth was reserved for the age of experiment2 and that she is alone to be apprehended in the infinite labyrinth of particulars8 ,that she is to be investigated with the corporeal senses1 and not with the powers of intellect2 and that the crucible1 the alembic1 and the air-pump1 are the only mediums of detection8 ,if this be the case1 truth is material1 and may be calcined1 distilled1 and rarefied1 like any other corporeal substance4 ,it is no longer eternal and immutable1 but perishable and fluctuating2 the phantastic subject of sensible inspection1 and not the steady and real object of the permanent energies of science4 ,shall we call this the age of philosophy1 in which talents are prostituted for sustenance1 and learning submits to the impudence of wealth8 ,shall we say that we have strengthened the cause of philosophy1 by demolishing her schools2 and increased her independence1 by enlarging the empire of commerce8 ,where shall we find the man1 who is at present reverenced for the profession of teaching speculative truth1 or indeed who teaches it at all8 ,or should we chance to meet with such an obsolete character1 shall we find him supported by the profession8 ,it is a well known fact1 that men formerly lived in the highest esteem by its propagation3 it is equally as notorious1 that a man at present would starve by such an attempt4 ,dare we assert1 that the reason of this difference must be ascribed to the greater liberality1 and more philosophical spirit of the present age8 ,shall we not rather say1 that the period1 in which these ancient heroes lived1 was the golden age of philosophy2--a period so different from the present1 as to appear fabulous on the comparison8 ,for mark the distinguishing characteristics of our inferiority4 ,the great object of ancient philosophy1 was an accurate speculation of principles and causes3 but that of the modern1 is a confused investigation of effects4 ,and if pursuits participate of the nature of their subjects1 and causes are more noble than effects1 the ancient philosophy must undoubtedly be more elevated than the modern4 ,again1 the object of the ,pythagorean and ,platonic philosophy was to make its possessors wise and virtuous2 and to elevate them above the common frailties and imperfections of degraded humanity2 and this end was happily accomplished in its votaries1 as their lives abundantly evince3 but the object of modern philosophy1 is a promotion of the conveniencies and refinements of life1 by enlarging the boundaries of traffic2 and the ,mathematical ,sciences are studied solely with a view to this enlargement4 ,the design of the ancient philosophy was to remove the causes of wonder1 by contemplating effects in their causes3 the grand object of the modern1 is to increase admiration1 by attempting to investigate causes through the infinity of particular effects4 ,so that philosophy1 as ,mr4 ,harris justly observes1 now ends where it formerly began4 ,for either there is no such thing as science1 or if its existence be admitted1 it can never be obtained by experimental enquiries2 as these must be liable to all the inaccuracy and imperfection of their material subjects4 ,in short1 the philosophy of ,pythagoras and ,plato will be found1 when impartially considered1 to contain every thing which can enlighten the mind1 improve the morals1 and exalt the character of man4 ,it is built on the steady basis of truth1 and will survive the wreck of ages4 ,its foundation is deep1 and its summit reaches the heavens4 ,it is a mighty rock1 which modern systems may assail1 like a raging sea2 but1 like stormy waves1 they will only be broken about its impenetrable sides4 ,to war against wisdom is folly2 for opposition in this case is the destruction of its author4 ,the moderns may1 indeed1 expect1 because their merit is raised by the present age1 above that of the ancients1 to appear as giants in the eyes of posterity2 but they will only verify the elegant observation of the poet#d1 that ,pygmies are ,pygmies still1 though perch'd on ,alps1 ,and ,pyramids are ,pyramids in vales4 ,a ,,dissertation ,,on ,,the ,,platonic ,,doctrine ,,of ,,ideas1 c4 ,,section ,i4 ,the ,platonic doctrine of ,ideas has been1 in all ages1 the derision of the vulgar1 and the admiration of the wise4 ,indeed1 if we consider that ideas are the most sublime objects of speculation1 and that their nature is no less bright in itself1 than difficult to investigate1 this opposition in the conduct of mankind will be natural and necessary2 for1 from our connection with a material nature1 our intellectual eye1 previous to the irradiations of science1 is as ill adapted to objects the most splendid of all1 8as the eyes of bats to the light of day#e40 ,and yet 7as ,i presume1 it will appear from the following discourse71 unless the existence of these lucid beings is admitted1 there can be no such thing as science2 nor1 indeed1 any genuine knowledge at all4 ,hence1 an enquiry concerning their nature and reality1 is highly proper1 as an introduction to the ensuing ,commentaries1 in which they are considered as the stable pillars of all truth1 and the prolific principles of the universe4 ,but previous to this enquiry1 it is proper to observe1 that ,plato was not the inventor1 though he was a strenuous asserter1 of ideas2 for1 in the ,sophista he affirms1 that ideas were the discovery of men who excelled in wisdom and piety1 and who contended for an invisible essence4 ,diogenes ,lartius1 indeed1 asserts1 that ,plato received the doctrine of ideas from ,epicharmus4 ,but ,epicharmus was not their inventor1 because ,pythagoras1 and others of still higher antiquity1 were well acquainted with ideas2 so that it may be affirmed1 with much greater truth1 that ,plato was instructed in their nature by ,philolaus his preceptor1 and the disciple of ,pythagoras4 ,for ,pythagoras1 after his mysterious manner1 signified ideas by numbers4 ,but1 prior to ,pythagoras1 ,orpheus was an asserter of ideas1 and called ,jupiter1 or the dimiurgus of the world1 8the idea of all things40 ,and1 according to ,syrianus1 the mundane sphere1 celebrated by ,empedocles1 is no other than the ideal world2 so that the doctrine of ideas is as ancient as that of wisdom itself4 ,but to begin with our enquiry3 in the first place1 without universals there can be no science2 for the flowing and perishing nature of particulars is perfectly foreign from that stability and duration which is requisite to objects of invariable truth4 ,neither is it possible1 that infinite individuals can exist without the subsistence of one cause endued with infinite power2 for all multitude must necessarily originate from one1 and must resemble its cause in as great a degree of perfection as its nature can admit2 by a diffused infinity1 shadowing forth that infinite power which subsists in indivisible union4 ,hence1 if this be the case1 and if infinite men1 horses1 and a multitude of other univocals1 are produced in an infinite time1 an unity of infinite power must be the source of each1 according to which they are generated in a terminated manner to infinity in the universe4 ,again1 all animals are transmuted from that which is in capacity 7i4 e4 seed71 into energy4 ,but if this be true1 it is requisite there should be some animal in the universe1 subsisting in ever-vital energy1 which may call forth that which is concealed in dormant capacity1 into perfect actuality4 ,thirdly1 the celestial orbs would not perpetually revolve in the same spaces1 and after the same manner1 unless one and the same universal number1 or idea1 ruled in each4 ,so1 likewise1 there is a natural number in every animal2 or those of the same species1 would not always 7when perfect7 be distinguished with the same invariable organs2 nor would they be subject to puberty and old age1 at the same time1 unless they were detained by the same measure of nature4 ,besides1 the participation of universals1 is evident in every sensible object4 ,thus1 the rational nature is united with every individual man4 ,thus1 animal subsists in a lion and a horse1 in a man and a dog4 ,and thus the pentad1 or number five1 is participated in the five fingers1 and the duad in the nostrils1 eyes1 hands1 and feet4 ,but since these do not subsist without a cause1 but are perfected by certain determinate natures1 it is necessary there should be an universal animal1 in the whole of nature1 separate from sensibles1 by means of which this sensible animal is generated4 ,and that there should subsist in nature a pentad1 through which the hands are always adorned with that number of extremities2 and a duad1 from which the two eyes and nostrils are derived4 ,but if nature does not possess these numbers from herself1 as she is not the first cause of all1 but derives them from another cause1 in the same manner as matter from nature1 it is necessary there should be universals and numbers prior to nature1 subsisting in far greater purity and perfection4 ,again1 we may demonstrate the existence of ideas as follows3 if the ,deity1 in fabricating the universe1 operated essentially 7and there is no other way in which we can conceive him to operate71 he must fabricate the universe1 an image of ,himself4 ,but1 if this be the case1 he contains in himself1 in the manner of an exemplar1 the causes of the universe2 and these causes are no other than ideas4 ,besides1 this consideration is not to be omitted1 that the perfect must necessarily antecede and preside over the imperfect2 unity over multitude2 the impartible over the partible2 and that which is perpetually the same1 over that which admits of variation and change4 ,from whence it may be inferred1 that things do not originate from baser natures1 but that their gradual processions end in these2 and that they begin from the most perfect1 best1 and most beautiful natures4 ,but let us pursue this reasoning more minutely1 as it affords the strongest arguments for the existence of ideas4 ,when the ,deity fabricated the various species of animals1 and bestowed on them the different senses1 it was doubtless with a view to the benefit of their possessors1 as he foresaw1 that without these1 the animal could neither provide for its own support1 nor defend itself from surrounding dangers4 ,but may we not enquire from whence this previous perception originated8 ,for it is not to be supposed1 that he first made animals destitute of senses1 and so1 being admonished by their sudden destruction1 afterwards assigned them to their nature4 ,shall we say1 this foreknowledge was the result of a reasoning process8 ,but then1 we again ask1 ,what were the principles of this ratiocination8 ,for if they originated from other reasonings1 it is necessary1 at length1 to arrive at something prior to these discursive operations1 on which they ultimately depend2 since all reasoning must be founded on indemonstrable principles4 ,was sense1 then1 or intellect the principle of this previous perception8 ,but1 sense1 in the present instance1 had not then a being1 for it could not exist prior to the animal nature3 it was1 therefore1 intellect4 ,but if intellect be the repository of certain propositions1 and the conclusion be science1 it must follow1 that there could not then be a consultation of any thing sensible4 ,for the principle and the conclusion must both depend on something intelligible4 ,besides1 may we not ask1 how such a habit of thought arose before the existence of a sensible nature6 ,it is absurd in the extreme1 to say from chance1 and to resolve it into a sudden volition of the ,deity1 is an assertion that may1 indeed1 satisfy vulgar minds1 but can by no means quiet the restless spirit of philosophical investigation4 ,since1 to suppose the cause of the universe1 actuated by sudden volitions1 is to place him on a level with the vilest natures1 and subject him to the irrational impulses of the brute4 ,hence we infer that the formation of animals1 and by the same arguments of the world1 was not the result of any reasoning process4 ,for1 indeed1 argument and foreknowledge cannot with propriety be attributed to the ,deity2 but when they are ascribed to him1 we must consider it as nothing more than an indication of his constituting particulars1 in a manner somewhat similar to the providence of a wise man1 in inferior concerns4 ,for1 in subordinate natures1 whose operations cannot take effect prior to enquiry1 reason is necessary1 on account of the inferiority of that power which precedes the reasoning energy4 ,in like manner1 foreknowledge is necessary1 because a power is wanting to its possessor1 which might render him superior to its use4 ,for foreknowledge is directed to this end1 that one particular circumstance may take place in preference to another4 ,but if it be requisite that every energy in the ,deity should be void of defect1 and if it is not lawful that any thing should be present with him1 which is not total and universal1 it is necessary that all things should be contained in every thing essential to the nature of the ,deity4 ,hence1 since even futurity is with him present1 there is nothing in him posterior2 but what is present in him becomes posterior1 by its participation in another4 ,if then futurity be present with the ,deity1 it is necessary it should be so present1 as if foreknown in a posterior nature2 that is1 in such a manner that nothing may be wanting to any being2 and that is1 lastly1 so that every thing may be complete4 ,besides1 reasoning cannot1 by any means1 belong to an eternal essence like the deity2 for if this be admitted1 he must be forgetful of his former operations4 ,and if1 in consequence of reasoning1 he produces more perfect natures afterwards1 his works could not be perfectly beautiful before3 but if they were beautiful before1 they must be co-existent with their cause1 i4e4 they must be eternally beautiful1 antecedent to the reasoning energy4 ,again1 if we suppose the supreme intellect1 the demiurgus of the world1 to operate by enquiry1 his energy could not be spontaneous1 and truly his own2 but his essence would be similar to that of the artificer1 who does not derive his productions from himself1 but procures them as something adventitious by learning and enquiry4 ,but if the universe was not formed by deliberation1 it must be co-existent with its cause1 and reside in his essence2 for if it be not co-existent there must have been some particular time1 in which its artificer determined on its production2 and this determination must have been the result of a reasoning process1 concluding that it would not be good to produce it before that particular time1 7from whence1 by the way1 we infer the eternity of the world47 ,and if the universe be co-existent with its author1 it must perpetually emanate from his nature1 and be dependent on it1 like the shadow on its forming substance4 ,but in this case1 its archetype must be contained in the essence of its author2 for every cause is that primarily1 which its effect is secondarily4 ,and hence we infer1 that if the sensible universe be replete with forms of every kind1 the exemplars of those forms1 must subsist in immaterial perfection1 in the artificer of the world4 ,if this sensible world1 then1 be formed according to the exemplar of that which is intelligible2 may we not say1 with the great ,plotinus1 that it is requisite universal animal should there primarily subsist in perfect vital energy1 containing all things in its omniform essence4 8,hence 7says he#f7 the heavens are there a divine animal1 replete with ideal stars4 ,earth too does not there subsist solitary1 but is much more vital than this corporeal earth1 for it is full of intellectual life4 ,the sea too is there1 and all water subsisting in life1 and an ever-abiding stream4 ,for how is it possible that any thing not vital1 can be the progeny of life itself8 ,he1 therefore1 who enquires from whence animals originate in the intelligible world1 might as well enquire from whence all life1 and soul1 and universal intellect1 arose4 ,for here there is nothing indigent nor defective1 but every thing is perfect and exuberant4 ,here they all flow from one fountain1 not as from a certain spirit1 or heat1 but as if from an universal quality1 possessing and preserving in itself1 all qualities2 such as sweetness1 accompanied with fragrance of smell1 the vigour of wine1 and the strength of all juices1 bright colours1 and whatever is perceived by the taste40 #c4 ,such then are the arguments which the ,platonic philosophy affords in defence of ideas2 the existence of which was so evident to ,plato1 that1 in the ,sophista1 he compares those who oppose the friends of ideas to the giants of old1 warring1 as it were1 on celestial souls1 and such as are engaged in sublime investigations4 ,let us now consider to what universals these lucid beings are confined2 since1 according to the ,pythagoreans and ,platonists1 there are not ideas of all universal conceptions4 8,for1 in the first place 7says ,syrianus#g71 there are no ideas of things evil and base1 because these subsist in nature rather by a privation and absence of ideas4 ,and1 on this account1 they are said to exist contrary to nature4 ,nor1 secondly1 of negations1 for these are destructive of the bound and limitation which is attributed to every thing from the unifying and comprehending nature of ideas2 and hence1 separation is rather the result of material infinity than of that which is formal or ideal4 ,nor again1 are there any ideas of things which at different times receive a variety of conditions4 ,for these participate of transmutation from a moveable cause1 but not from the immoveable and stable illustration of ideas4 ,nor again of parts1 such as the hand1 head1 fingers1 and the like4 ,for the causes of things existing entire1 produce whole species and forms2 not divided about the parts of these1 like the reasons of nature4 ,but neither did these wise men place in intellect the determinate causes of accidents in bodies1 such as sweetness and whiteness4 ,for they considered that natural reasons were sufficient for the production of accidents4 ,nor again1 of composites1 as of a wise man4 ,for since ideas are simple1 they preside over the simple essence of every thing4 ,but the composition and division of things is the business of our intellect2 ideas1 at the same time1 and that intellection which is co-ordinate to ideas1 being exempt from all these1 on account of superlative simplicity4 ,neither1 therefore1 must we establish ideas of things generated from dissimilars1 such as mules2 nor of fruit produced by engrafting from different trees4 ,for all these have a posterior and adventitious generation1 and are not the work of nature alone1 nor of nature proceeding according to her own reasons1 but1 as it were1 compelled to labour contrary to her own determinations4 ,hence it is manifest1 that all art1 which imitates nature1 and alone ministers to the use of mortal life1 is separated from the cause of ideas4 ,but neither are the works which1 depending on the purpose of the soul1 are perfected by a concourse of many causes1 and which we are accustomed to call the operations of fortune1 to be conjoined to the cause of ideas4 ,for things which are there perfected1 are eternal1 and subsist perpetually the same1 free from the nature of contingent events4 ,it remains1 therefore1 that ideas must be confined to universal and perfect essences1 and to whatever confers to their natural disposition2 as for instance1 to man1 and every thing perfective of man1 such as wisdom and virtue4 ,for ideas existing as the generative and energetic causes of the perfection of every thing1 distribute being to essences1 and convert them to the inexhaustible plenitude of their own omniform natures40 #d4 ,but let us now consider the nature of numbers2 for as every form is a number1 according to the ,pythagoreans#h1 a speculation of this kind must afford no small light to the arduous investigation of ideas4 ,will it not1 therefore1 be proper1 in the first place1 to enquire1 with the great ,plotinus#i1 whether multitude is not a departure and distance from one1 so that infinity itself is a separation from unity in the extreme1 because it is no other than innumerable multitude2 that on this account it becomes evil2 and that we contract a similar nature when departing from intellectual unity1 we are divided by sensible multitude8 ,for a being then properly becomes many1 when no longer able to remain collected in itself1 the same1 it is diffused abroad1 and thus1 being dispersed1 is variously extended2 so that when1 by diffusion1 it is absolutely deprived of unity1 it becomes perfect multitude1 destitute of that universal cement1 which unites one part with another4 ,but whenever the conciliating one is present1 then that which was scattered and diffused1 becoming permanent by its bounding power1 passes into magnitude4 ,but if any one should deny the subsistence of unity1 asserting that one is no where to be found1 which is not some particular one2 and should hence affirm1 that what is called one abstractedly1 is only a certain affection of the soul towards any being2 we ask1 what prohibits the appellation of essence1 from being nothing more than an affection of the soul1 and consequently the existence of being1 a delusion8 ,for we predicate unity of particulars with as great propriety as being4 ,i am well aware1 that philosophers of the present day will answer1 that we have an evident proof of the reality of being1 from its agitating the soul1 and becoming apparent in the phantasy3 to which we reply1 that in like manner1 the soul is agitated1 and the imagination influenced about the one4 ,for every individual as much excites the perception of one1 as of being4 ,besides1 it is necessary to enquire whether we behold this passion and conception of the soul1 as one or multitude4 ,and again1 when we say not one1 we do not then possess one from the thing itself2 for we say that one is not contained in that individual4 ,and hence we must possess one in our own nature1 and this must reside in the soul1 separate from that which is denominated some particular one4 ,but here it may be objected1 that the one we possess is received from externals1 and is nothing more than a conception of the mind1 produced by the thing itself4 ,for it will be said1 that as multitude is nothing besides a number of individuals1 which are called many1 so one is nothing besides one thing2 and is formed by thought separating that one particular from others4 ,to this we reply as follows3 ,how can it be consonant to reason to suppose that the conception of one arises from the sensation of some one particular subject8 ,for one particular man1 who is discerned by sense1 is by no means the same with one itself1 since1 if this were the case1 thought could never predicate one of that which is not a man4 ,besides1 as cogitation1 on beholding the different positions of things1 affirms that this is here or there1 so when it perceives an individual1 pronounces one2 for that passion is not vain1 nor does it assert one of a non-entity4 ,nor must we think it predicates ones1 because this individual is different from another2 for when cogitation affirms such a thing is this1 and not another1 it declares1 in the mean time1 that the other is one4 ,likewise when it affirms that any thing is this alone1 it then declares1 that what is alone is one3 on which account1 it predicates one1 prior to alone4 ,besides1 if there be multitude1 it is necessary that one should antecede2 since when it predicates many1 it pronounces more than one4 ,and when it affirms that an army contains a multitude of men1 it conceives the soldiers reduced to one order4 ,for thought1 indeed1 does not permit multitude to remain perfect multitude1 destitute of the conciliating power of unity2 in which very circumstance1 the subsistence of one is evinced2 for acutely and swiftly perceiving the one which results from order1 it reduces the nature of the many into one4 ,besides1 we affirm that a house and an army are each one1 but that a house is more one than an army1 on account of the continuity of its parts4 ,if therefore1 one is contained more in that which is continued than in that which is discrete1 and still more in what is perfectly indivisible1 it is evident that the one is a certain nature1 and has a real being4 ,for it is impossible that the more and the less should take place among things which have no subsistence4 ,if then it be not possible to understand any thing without one or two1 or some other number1 it is by no means proper to deny existence to that1 without which we cannot comprehend the existence or properties of any being3 but it is requisite that nature should antecede all discourse1 and intelligence1 which is every where necessary to their existence4 ,again1 if unity has no real subsistence1 and is nothing more than a name or conception of the mind1 it may be destroyed without the destruction of its subject4 ,the unity1 therefore1 of a house may be taken away1 without the ruin of a house4 ,but if a house is nothing more than certain materials1 reduced into one form1 this is impossible4 ,and1 on the contrary1 the alteration of that subject1 of which unity is predicated1 can make no real alteration in unity 7on this hypothesis7 any more than the death of a man can affect his name4 ,when1 therefore1 a body1 of which one was predicated1 is divided into a multitude of parts1 there is no real alteration made in the unity of the body1 because unity is nothing more than a name4 ,it was in consequence of this reasoning1 and perceiving that unity was participated by every being1 that the ,pythagoreans placed a super-essential one at the top of the universe1 intelligibly abstracted from all beings in simplicity and excellence of nature4 ,for they considered1 that unless there was a self-subsisting one in all things1 there could neither be universals nor particulars4 ,not the first1 because they are by nature one and many4 ,but it is requisite that the one itself1 should preside over that which is not one alone4 ,nor again1 the second1 because they are many and one1 7that is1 they participate more of multitude than unity1 and their nature is determined more by the many than the one47 ,and because of things in participation1 unless an unparticipated one is added1 there can be no cause of union to beings2 in the same manner as the cause of essence to beings1 is taken away by those who deny that being itself1 is the principle of all essence4 ,for as the good itself1 is the one principle of good to the universe1 and is nothing besides good2 and as a self-motive nature1 which is nothing besides self-motion1 is the cause of motion to all things2 so all things proceed from being itself1 and all united natures receive their union from the one1 abstracted from all things4 ,hence 7such is the absolute dominion of unity71 continued quantities would have no existence without its participation2 for when they are divided1 so far as they lose unity1 they change their being into some other form4 ,hence1 the bodies of plants or animals1 which are each of them one1 when they fly from unity1 and are dissipated into multitude1 immediately lose the essence they formerly possessed1 and become something else2 which new state of being they likewise possess so far as they are one4 ,add too1 that health then flourishes in the corporeal frame1 when the body is conciliated into one2 then beauty flourishes1 when the power of one connects the members into proportion and consent2 and then virtue reigns in the soul1 when the soul is reduced into one similitude with that which is divine4 #e4 ,but let us now investigate the nature of numbers4 ,all number1 according to the ,pythagoreans1 originates from unity and the indefinite duad2 the first having the relation of form1 and the second1 that of matter to all the orders of numbers4 ,but they likewise divided number into two kinds1 essential and monadic4 ,the essential number they considered as first subsisting in the intelligible world1 together with being1 and from thence distributed into all the various gradations of forms4 ,but the monadic1 or that which is composed from certain units1 they justly considered as nothing more than the image of essential number4 ,and with respect to the numbers which the human soul participates1 these from its imperfect condition have a middle subsistence2 i4 e4 they exist in a vital1 gnostic1 and speculative1 but not in an operative manner4 ,hence1 when receiving one thing with another1 we affirm1 that they are two1 as a dog and a man1 or two men2 or when we compute more than two1 as ten1 and say that there is a decad of men1 this number is not essential to the two or ten individuals1 nor is it to be conceived as subsisting in sensible natures2 but it is purely quantity4 ,but when we distribute this ten1 into units1 we produce the principle of quantity1 and generate a subject in opinion#aj1 capable of participating the essential decad of our soul4 ,but when1 considering man in himself1 we affirm that he is a certain number1 as the duad1 composed of animal and rational1 we do not observe one mode in this predication2 but so far as by a discursive operation of the soul1 we numerate1 we effect a particular quantum2 but so far as the subjects are two1 and at the same time both one 7since one fills the essence of both1 and in both unity is contained71 we pronounce another1 and an essential number3 and this duad is not of a posterior origin1 nor alone signifies a certain quantity1 external to the subject1 but a duad subsisting in the essence of man1 and containing his nature4 ,for here we do not produce a number by a discursive operation1 while we pursue essential natures4 ,but when we number any ten things1 which are not connected by any conciliating unity1 like a choir1 or an army1 then this decad1 which we predicate of the ten particulars1 subsists alone in our numerating soul1 which renders the ten individuals in opinion1 a definite quantum4 ,but in a choir1 or an army1 essential number is participated exclusive of that which subsists in our soul4 ,and if it be enquired how number subsists in the human soul1 we must say1 that the soul1 by her self-moving energies1 procreates number1 while she numerates1 and by this energy1 causes the existence of quantity2 in the same manner as in walking1 we give rise to a certain motion4 ,thus1 monadic number1 or a collection of units of various kinds1 subsists in opinion1 in a manner correspondent to that of geometrical figures2 and by this means participates the essential number of the soul4 ,for as a triangular figure in the phantasy1 is the recipient of a triangular nature1 or of triangle itself2 so every three units in opinion1 receive the essential triad of the soul1 and1 by this means1 form a definite quantum4 ,in short1 as in every being we may discern the resemblances of matter and form1 so in the pentad1 or any other number1 the five units1 which are the subject of participation1 and the quantity of the number1 originate from the duad2 but the form1 that is the pentad itself1 from unity4 ,for every form is an unity1 which unites its subject quantity1 and connects it with its ideal species4 ,it is1 therefore1 requisite to understand1 that the two principles of mathematical numbers are resident in our souls1 with which every mathematical number is co-existent2 ,i mean unity1 comprehending in itself all the forms of numbers1 and which corresponds to unity in intellectual natures2 and the duad1 endued with a generative power1 of a formless nature1 and of infinite virtue2 and which is called boundless1 on account of its being the image of never-failing and intelligible duality4 ,hence1 the unity of the soul1 with a never-ceasing energy1 continually distinguishes and forms all the orderly processions of her numbers1 suffers no vacuum to intervene1 and leaves no quantity formless and innumerable4 ,hence too1 no essential number of the soul1 as for instance1 the pentad1 is composed from substance and accident1 as a white man2 nor from genus and difference1 as man from animal and biped2 nor again1 from five unities mutually touching each other1 like a bundle of wood2 nor from things mixt1 like water and wine1 nor from things subsisting by position1 in the manner that stones compose a house2 nor lastly1 does it subsist like things numerable2 for it is not because they are composed from indivisible units1 that they possess any thing besides units4 ,for many points are indivisible1 yet quantity is not produced on this account2 but because they participate of two natures1 the one corresponding to matter1 and the other to form4 ,lastly1 it is not proper to say1 that the number seven 7and so of any other number71 is composed from the triad and the tetrad2 for units1 indeed1 composed with units1 form a subject adapted to the reception of the heptad1 or the ideal and essential number seven2 but the definite numerical quantity seven1 is formed from so many units1 and the ideal heptad4 ,hence1 as the soul of the shipwright gives form to the timber1 from her inherent art2 so the numerative soul1 from the unity endued with the relation of a principle which she possesses1 gives form and subsistence to all her inherent numbers4 ,but there is this difference between the two1 that the shipwright's art is not essential to our nature1 and requires manual operation1 because it is conversant with sensible matter2 but the numerative art is essentially inherent in the soul1 and is therefore present with all men1 and possesses an intellectual matter1 which it easily forms without the assistance of time4 ,and this1 perhaps1 is what deceives many1 who think that the heptad is nothing more than seven units4 ,for the imagination of the vulgar1 unless it first perceives a thing destitute of ornament1 and afterwards the operations of the adorning artificer supervening its nature2 and lastly1 beholds the thing perfect1 and invested with form1 cannot be persuaded that it possesses two natures1 the one formless1 but the other endued with an energetic and forming power4 ,and here it is necessary to observe1 that though unity is the form of all arithmetical forms1 yet it is chiefly the form of the decad4 ,for what unity is simply to all the series of numbers1 that the decad is to the following hundreds1 thousands1 and millions2 from whence1 according to a secondary progression1 it is denominated unity4 ,as intellect1 therefore1 is the form of all things1 but especially of the soul1 so unity1 though it is the idea of all numbers1 yet especially of the decad4 ,but the reason why the ,pythagoreans extended ideal numbers no farther than ten1 is because this number is the ultimate perfection of beings1 containing all things in its omniform nature4 ,for all proportion subsists within the number ten2 the arithmetical in a natural progression of numbers from unity2 the geometrical in the numbers #a1 #b1 #d1 and #a1 #c1 #i1 and the harmonical in the numbers #b1 #c1 #f1 and #c1 #d1 #f4 ,and since the causes of all things are contained in numbers1 as far as to the decad#aa1 it is superfluous to suppose exemplars of the following numbers4 ,if it should be asked in what manner we must conceive number as subsisting in the intelligible world1 we answer1 with the great ,plotinus1 that we must conceive it as subsisting in being itself1 with a power of impelling it to the production of multitude4 8,hence 7says he1 ,ennead vi4 lib4 vi47 number is either the essence or the energy of being1 and animal itself1 and intellect is number4 ,but1 perhaps1 we must call being1 number united 7 71 but beings1 number evolved1 or unfolded2 7 7 intellect1 number moving in itself2 7 7 and lastly1 animal1 number comprehending 7 487 ,it was in consequence of this reasoning1 that the ,pythagoreans called ideas numbers2 because the gradual evolution of these from ineffable unity1 produced all the beautiful variety of forms4 ,their exalted conceptions of numbers1 likewise1 originated from the same sublime theory4 ,hence1 #ab,pythagoras1 in the sacred discourse1 calls number 8the ruler of forms and ideas40 ,but #ac,philolaus1 8the commanding and self-begotten container of the eternal duration of mundane concerns40 ,and #ad;,hippasus1 and all those who were called 7or such as were yet under the probation of the quinquennial silence71 8the first exemplar of the mundane fabric1 and the judiciary instrument of its artificer40 #f4 ,and here ,i cannot but take notice1 with regret1 of the very unphilosophical mistake committed by that great mathematician ,dr4 ,barrow#ae3 ,i say1 with regret1 on account of the extraordinary obligations ,i am under to his writings1 for my proficiency 7whatever it may be7 in mathematical learning4 ,but respect must yield to the truth4 8,unity1 says he1 is not indivisible4 7,for how ex4 gr4 can #b_/f added to #d_/f be equal to unity1 if unity be indivisible and incomposed1 and represent a point7 but rather only unity is properly divisible1 and numbers arise from the division of unity40 ,here the ,doctor evidently confounds sensible units1 which are the subjects of vulgar practical arithmetic1 with those units which are the objects of science4 ,every individual sensible object1 is indeed an unit1 so far as it participates the connecting and conciliating power of an immaterial one3 but the unity which stands at the top of speculative arithmetic1 is perfectly indivisible1 or arithmetic would cease to be a science4 ,the truth of this is evident from ,euclid's definition3 8,unity 7says he7 is that according to which each of the things which are1 is called one40 ,but if unity be a composite1 the definition is false2 since a composite1 or a certain multitude1 can never be the cause of unity1 but the contrary4 ,and that this immaterial one subsists in sensible natures1 has1 ,i hope1 been sufficiently proved in the preceding part of this discourse4 ,but the ,platonic ,theo#af of ,smyrna1 fully establishes the indivisibility of unity1 as follows3 8,unity is terminating quantity1 the principle and element of numbers1 which remains undiminished by the most immense multitude of subtractions1 and being deprived of all number1 continues firm and fixt1 because it is impossible for division to proceed beyond the bound of unity4 ,thus1 if we separate any one corporeal substance into parts1 the one again becomes many2 and by subtracting the several parts1 we end in one part2 and from this remaining part1 again divided1 arises multitude2 and by taking away every part1 we again arrive at one4 ,so that one1 considered as one1 is incapable of diminution1 and perfectly indivisible4 ,on the contrary1 every number is diminished by division1 and is separated into parts less than itself2 as the number #f into #c and #c1 or into #d and #b1 or into #e and #a4 ,but unity in sensible particulars1 if divided1 is diminished after the manner of body1 and by section is distributed into parts less than itself3 but it receives increase after the manner of number2 for instead of the one1 multitude is produced4 ,in this sense1 therefore1 is unity indivisible2 for nothing is divided into parts greater than itself4 ,but that which is cut into parts greater than the whole1 and into parts equal to the whole1 is divided as number4 ,thus1 for instance1 if any one sensible body is divided into six parts1 #a1 #a1 #a1 #a1 #a1 #a1 these shall be equal to the whole2 but by a section into #d and #b1 it is divided into parts greater than the whole1 considered as one2 for #d and #b considered as numbers1 exceed unity1 and the body was supposed to be one4 ,unity1 therefore1 as number is perfectly indivisible4 ,but unity is called by the ,greek word 1 only1 or alone1 either because it remains immoveable1 and does not desert itself1 nor surpass the bounds of its nature 7for it remains the same1 however multiplied into itself1 through an infinite progression7 or because it is placed separate and apart from the multitude of other numbers1 it is denominated the monad1 or one40 ,in consequence of this very mistaken hypothesis1 which opposes not only all the wisdom of antiquity1 but the sublimest truths1 the ,doctor asserts1 that an arithmetical cypher is the principle of numbers2 and that it is analogous to a point in geometry4 ,just as if a cypher1 which is nothing more than a mark expressive by its position with numbers1 of a certain quantity1 had a real existence1 and was productive of number3 when1 at the same time1 any other arbitrary character would serve the same purposes1 if applied in a similar manner4 ,it must surely afflict every thinking mind1 to see how dreadfully the mechanical system of philosophy1 which has been so long in fashion1 enslaves and perverts the minds of its votaries2 for there cannot1 ,i think1 be a more egregious instance of its fatal tendency1 than the present1 in which nothing is considered as the foundation of that noble science1 arithmetic2 which was deservedly placed by the ancients1 in the first rank of the mathematical disciplines4 ,such a foundation1 indeed1 may be proper to the mechanical philosophy1 but is very ill adapted to support the solid fabric of the arithmetical science4 ,but let us attend to the arguments of this most learned man1 in defence of so strange an assertion1 8,a cypher1 or arithmetical nothing 7says he7 is really the bound of every number coming between it and the numbers next following1 but not as a part4 ,a cypher being added to1 or taken from a number1 does neither increase nor diminish it2 from it is taken the beginning of computation1 while itself is not computed2 and it bears a manifest relation to the principal properties of a geometrical point40 ,but in what manner are we to conceive the nothing which intervenes between any two numbers1 to be their term or boundary8 ,for ,euclid defines a term to be the extremity of any thing2 implying by the extremity1 something belonging to that of which it is the bound4 ,but how can a cypher1 or nothing1 in any respect belong to number1 or something8 ,for if nothing be a boundary1 merely from its intervention1 a point existing between any two disjoined lines1 though at the greatest distance from each1 must be their common boundary1 which is evidently absurd4 ,besides1 what relation does it bear to a point1 which is endued with a generative power1 by its flux forming the simple extension of a line1 and1 at the same time1 every where limiting its progression1 and subsisting in infinite capacity in its every part8 ,where are the real and divine properties to be found in an arithmetical nothing1 which ,proclus1 in the following ,commentaries1 exhibits in a point8 ,and how can computation originate from a mere non-entity8 ,but a little consideration will convince us1 that this ,saracen1 or ,indian cypher1 is nothing more than an arbitrary character1 invented for the purpose of facilitating computation4 ,for1 suppose the letter 7a7 to be placed in its stead1 and to signify1 when connected with the mark for unity1 ten1 or ten multiplied by one2 when connected with the mark for two1 ten multiplied by two1 and so on4 ,and again1 when placed twice after unity1 let it express the second power of ten1 or one hundred1 in this manner1 a a2 when thrice connected1 one thousand1 or the third power of ten1 and so on3 shall we say1 in consequence of this1 that 7a7 is the bound of numbers1 and the principle of arithmetic8 ,or1 shall we not rather say1 that it is an arbitrary symbol1 like any other algebraic character1 having no real connection with numbers1 and depending1 for its existence and application1 entirely on the will of its inventor4 ,but this opinion is too absurd to need any farther refutation4 #g4 ,it may here1 perhaps1 be expected1 that ,i should explain how1 in the language of ,syrianus#ag1 8divine number proceeds from the immortal retreats of unity1 until it arrives at the divine tetrad#ah20 and that ,i should unfold the properties of the tetractys1 according to the ,pythagoreans2 but an undertaking of this kind1 would not only far exceed the limits of this dissertation1 but1 perhaps1 in the present age1 might be justly deemed1 by the lovers of wisdom1 a prostitution and profanation of the most exalted truths4 ,enough1 ,i hope1 has been said to excite the curiosity1 and rouse the attention of the thinking and liberal part of mankind2 and those who understand what is here briefly delivered1 may apply themselves1 with advantage1 to ,proclus on ,plato's ,theology1 where they will find all the mysteries of numbers unravelled2 and to the works of the great ,plotinus1 who will lead them into the penetralia of the most recondite wisdom4 ,but1 in perusing the works of these great men1 the reader must not expect to find the sublimest truths explained in a familiar manner1 and adapted1 like many modern publications1 to the meanest capacities4 ,for this1 indeed1 is impossible to be effected4 8,mankind 7says ,petvin#ai71 are not to be made any more truly knowing than happy by another's understanding4--,there is no man can at once convey light in the higher subjects1 to another man's understanding4 ,it must come into the mind from its own motions1 within itself3 and the grand art of philosophy1 is to set the mind a-going2 and1 even when we think nothing of it1 to assist it in its labour40 ,after which he observes1 that 8the ancients never attempt to lead us into knowledge1 by a continued chain of reasoning2 on the contrary1 they write in such a manner1 as to force us to think for ourselves40 ,and1 previous to this1 he remarks1 8that there are certain truths acquired by a long exercise of reason1 both in particular1 and likewise in those subjects that are most general1 as much1 perhaps1 out of the reach of the greatest mathematician1 as ,sir ,isaac ,newton's speculations are above the capacity of some that are now called mathematicians40 ,the truth of this observation is sufficiently evinced1 in ,plato's definition of a philosopher 7in his ,sophista71 8,the philosopher 7says he7 is the man who sufficiently sees one idea every way extended through many1 every one of them lying apart2 and many ideas different from one another1 externally comprehended under one4--,and farther1 one idea1 throughout all manys1 wrapt up in one2 and many ideas1 every way separate or discreet4 ,this is to have the knowledge to discern how ideas1 as they are general1 agree and disagree40 ,now1 he who thinks that a perception of this kind may be acquired by barely reading an accurate discourse on the nature of ideas1 composed in intelligible terms1 without1 at the same time1 employing a long course of profound meditation1 and patient thought1 knows but little the difficulty of the task1 and until he changes his opinion will never be the wiser4 ,but the folly and presumption of men1 with respect to this sublime philosophy1 is really unpardonable2 for there are very few who conceive that much previous instruction is requisite to its acquisition2 but almost every man decides peremptorily on the most abstract speculations1 and reckons himself sufficient for the most profound investigations4 ,in the sciences and arts they are willing to proceed to perfection by gradual advances2 but they consider philosophy as easy1 of instant access1 and hastily approach to her embraces with an assured confidence of success4 ,though1 like unhappy ,ixion1 through their presumption1 instead of a goddess1 they grasp nothing but an empty cloud4 ,plato was so sensible of this truth1 that1 in his seventh epistle to ,dion1 he expressly affirms1 that he neither has written1 nor ever will write explicitly concerning these sublime speculations2 8,for a thing of this kind 7says he7 cannot be expressed by words1 like other disciplines1 but by a lasting familiarity1 and conjunction of life1 with this divine object1 a bright light#bj on a sudden1 as it were leaping from a fire1 will illuminate the soul1 and there preserve and nourish its splendor4 ,he adds1 that a publication of such concerns1 is alone useful to a few of mankind1 who from some small vestiges previously demonstrated1 are sufficiently sagacious to their invention4 ,but it will fill others partly with a base contempt1 and partly with a rash and vain confidence1 as if they had now learned some very excellent things40 ,he then subjoins the following instance of the difficulty attending such an undertaking3 8,there are three things 7says he71 from which science must necessarily be produced2 but the fourth is science itself4 ,and it is requisite to establish the fifth as that which is the object of knowledge1 and has a true existence4 ,one of these is the name of a thing2 the second its definition2 the third the resemblance2 the fourth science4 ,now take each of these1 desiring to learn what we have lately asserted1 and think concerning them all1 in a similar manner4 ,a circle is called something1 whose name we have just expressed4 ,after this follows its definition1 composed from nouns and verbs4 ,for that which every where is equally distant from the extremes to the middle1 is the definition of that which we signify by the name of a round1 and a circumference1 and a circle4 ,but the third is the circle which may be painted1 or blotted out1 which may be made by a wheel1 or destroyed4 ,none of which affections1 the circle itself1 which each of these respects1 suffers1 as being of a different nature4 ,but the fourth is science1 and intellect1 and true opinion about these4 ,and this again must be established as one whole1 which neither subsists in voice1 nor in corporeal figures1 but in intellect and intelligence4 ,it is therefore manifest1 that this fourth is different from the nature itself of the circle1 and again different from the three we have previously mentioned4 ,but among the number of these1 intellect1 by its relation and similitude1 proximately adheres to the fifth1 while the rest are more remote from its nature4 ,the same may likewise be affirmed of a straight and crooked figure1 of colour1 and of the good1 the beautiful1 and the just4 ,and again1 of every body1 whether fashioned by the hand1 or the work of nature1 whether fire or water1 and the rest of this kind2 likewise of every animal1 and the manners of animals2 and of all actions and passions4 ,for unless1 among these1 some one1 after a manner1 receives that fourth1 he will never perfectly participate the science about the fifth40 ,he then proceeds to shew in what respect each of the preceding four are different from the fifth4 8,every circle 7says he7 which by the hands of men is either painted1 or fashioned by a wheel1 is plainly contrary to our fifth4 ,for it every where participates of the right-line4 ,but we must affirm1 that the circle itself has neither more nor less of any thing whatever2 that is1 it possesses in itself1 nothing of a contrary nature4 ,besides1 none of these are endued with any stability of name4 ,for nothing hinders our applying the appellation of straight to that which we now denominate round1 and calling the straight by the denomination of the round2 nor will there be any less stability in these1 when their names are changed into the contrary4 ,the same reasoning is likewise true of definition1 since it is composed from nouns and verbs1 which possess no stability4 ,and in a variety of ways1 it may be proved1 that no one of these four is certain and firm40 ,now1 this fifth division of ,plato's entirely respects ideas1 considered as flourishing in intellect2 by a conjunction with which1 we acquire true intelligence1 and the perfection of human knowledge4 ,the first three of the preceding are obnoxious to various mutations2 the fourth less2 but the last is perfectly stable and invariable4 ,the three first are rather conversant about the qualities of things1 about the image and shadow2 the fourth raises us to the participation of truth2 but the fifth to truth itself1 and permanent essence4 ,in the first degrees almost all are conversant2 in the fourth a few2 in the fifth1 all the gods1 but a very small part of mankind1 as it is asserted in the ,timus4 ,the four first may be known1 indeed1 without the fifth1 confusedly2 but from the knowledge of the fifth they become perfectly manifest1 as effects from the knowledge of their cause4 ,but we cannot1 by any means1 attain to the apprehension of the fifth1 unless we have been first accurately conversant with the rest2 for from our imperfect condition we are compelled to rise from difference to identity1 from multitude to unity1 and from shadow to substance4 ,while we investigate the knowledge of things1 if we are alone desirous to apprehend their resemblance 7which is the case with the multitude7 we shall be placed in the third degree1 and may easily acquire the object of our pursuit4 ,but if we should fortunately possess the true philosophical genius1 which is rare in the extreme1 and aspiring to the fifth degree1 should1 by a happy event1 attain to its conjunction1 though such a contact is clearer and more certain than all knowledge2 yet it is difficult to express it in words1 and to manifest it to others4 ,and the reason of this is obvious3 first1 because words are wanting1 which exactly correspond to the essence of a thing1 since these are only the symbols of shadows4 ,secondly1 because we speak with those1 who are alone conversant with shadows1 and are on this account derided by them1 when they find that our fifth does not1 by any means1 accord with material resemblances1 which they consider as the only realities4 #h4 ,and here a question very naturally presents itself for our solution1 whether the soul1 while united with the body1 is able to perceive ideas1 without the assistance of the phantasy1 ,for it seems difficult to apprehend how the soul1 thus depressed and weighed down with the terrene mass1 should be able to raise herself to the supernal light of ideas1 and become united with their refulgence4 ,the opinion of the ,peripatetics is well known1 that some phantasm must always accompany intelligence2 but this is denied by the ,platonists1 and ,i think with great reason4 ,for the operations of intellect are not dependent on the phantasy1 though the perceptions of the latter proceed from the energies of the former4 ,besides1 as ,plotinus beautifully observes1 our most vigorous energies are accompanied with the least animadversion2 and there is no absurdity in supposing that by increasing the force of intellectual energy1 we may speculate free from all imagination2 since the phantasms attending our conceptions1 became weak in proportion as the intellectual sight increases in vigour4 ,on this account1 the ,platonists affirm1 that the moral virtues free us from the vehemence of perturbations2 but the contemplative from imagination1 and the senses4 ,hence too1 the sciences may be called living waters2 in which the wings of the soul being dipt1 her feathers1 which were either separated or broken by her lapse into body1 are repaired1 and restored to a resemblance of their former perfection4 ,for the wings are the powers of the soul1 leading to intelligibles3 but the feathers are as well the natural instincts to good and truth1 as reasons inserted in the soul2 which either fall off1 or are broken by her descent into body1 and conjunction with its ruinous bonds4 ,but these are repaired and invigorated by the sciences1 which1 like living streams1 flowing from the fountains of ideas1 restore life and perfection to the soul4 ,hence ,plato1 in the ,phdrus1 asserts that these wings of the soul are increased by every thing which confers to supernal elevation2 as beauty1 wisdom1 and the like2 and by a convenient metaphor1 in the same dialogue1 he considers the chariot of the souls lives1 her charioteer1 and the horses by which her car is drawn2 and lastly1 every thing which contributes to the elevation of the soul1 and her conjunction with intellect and ideas4 ,we may therefore conclude1 that this conjunction is possible to be effected1 though it is rarely obtained2 and that it is a flight too arduous and sacred for the groveling and sordid2 a splendor too bright for the sensible eye2 and a contact too ineffable to be described by the unstable composition of words4 ,but ,i cannot conclude this section1 without soliciting the reader's attention to a comparison of the difference between the ancient philosophy1 and that invented by ,mr4 ,locke1 and the moderns4 ,according to ,mr4 ,locke's system ideas are formed from sensible particulars1 by a kind of mechanical operation2 so that truth is something by its nature1 posterior to sensation1 and entirely dependent on it for existence4 ,according to ,plato1 ideas are eternal and immaterial beings1 the originals of all sensible forms1 and the fountains of all evidence and truth2 so that on this system truth ranks among the first1 and not in the last of things2 and would still retain its nature1 though the corporeal senses were no more4 ,according to ,mr4 ,locke1 the soul is a mere rasa tabula1 an empty recipient1 a mechanical blank4 ,according to ,plato1 she is an ever-written tablet1 a plenitude of forms1 a vital and intellectual energy4 ,on the former system1 she is on a level with the most degraded natures1 the receptacle of material species1 and the spectator of delusion and non-entity#ba4 ,hence1 her energies are nothing but somnolent perceptions1 and encumbered cogitations2 for all her knowledge terminates in sense1 and her science in passion4 ,like a man between sleeping and waking1 her visions are turbid and confused1 and the phantoms of a material night1 continually glide before her drowsy eye4 ,but on the latter system1 the soul is the connecting medium of an intelligible and sensible nature1 the bright repository of all middle forms1 and the vigilant eye of all cogitative reasons4 ,hence she is capable of rousing herself from the sleep of a corporeal life1 and emerging from this dark ,cimmerian land1 into the regions of light and reality4 ,at first1 indeed1 before she is excited by science1 she is oppressed with lethargy1 and clouded with oblivion2 but in proportion as learning and enquiry stimulate her dormant powers1 she wakens from the dreams of ignorance1 and opens her eye to the irradiations of wisdom4 ,on ,mr4 ,locke's system1 the principles of science and sense are the same1 for the energies of both originate from material forms1 on which they are continually employed4 ,hence1 science is subject to the flowing and perishable nature of particulars2 and if body and its attributes were destroyed1 would be nothing but a name4 ,but on the system of ,plato1 they differ as much as delusion and reality2 for here the vital1 permanent1 and lucid nature of ideas is the fountain of science2 and the inert1 unstable1 and obscure nature of sensible objects1 the source of sensation4 ,on ,mr4 ,locke's system1 body may be modified into thought1 and become an intelligent creature2 it may be subtilized into life1 and shrink1 by its exility1 into intellect4 ,on that of ,plato1 body can never alter its nature by modification1 however1 it may be rarefied and refined1 varied by the transposition of its parts1 or tortured by the hand of experiment4 ,in short1 the two systems may be aptly represented by the two sections of a line1 in ,plato's ,republic4 ,in the ancient1 you have truth itself1 and whatever participates of the brightest evidence and reality3 in the modern1 ignorance1 and whatever belongs to obscurity and shadow4 ,the former fills the soul with intelligible light1 breaks her lethargic fetters1 and elevates her to the principle of things2 the latter clouds the intellectual eye of the soul1 by increasing her oblivion1 strengthens her corporeal bands1 and hurries her downwards into the dark labyrinths of matter4 ,nor is it wonderful there should be so great a difference between the two systems1 and so much in favour of the ancients1 if we consider the great advantages these ancients possessed over the moderns in every thing which contributes to the advancement of philosophy4 ,for1 in the first place1 they lived in an age when abstract investigations were in the greatest request1 and the professors of such pursuits in the highest estimation4 ,besides this1 they united the most exalted abilities with the most unwearied attention and obstinate perseverance2 they devoted their whole lives to the search of truth2 and relinquished every thing which might be an obstacle to its acquisition4 ,we may add1 likewise1 the advantages of a language extremely philosophical2 and a freedom from the toil of learning any tongue but their own4 ,now the reverse of all this is the portion of the moderns3 for in the present age1 abstract speculations are ridiculed2 and its professors despised4 ,the pursuit of truth is considered as perfectly consistent with ordinary avocations1 and is rather prosecuted as a relief from the toils of business than as a thing desirable for its own sake1 and of the greatest dignity and worth4 ,hence1 a few years desultory application at a college1 where language is one of the first objects of attention1 qualifies a modern for philosophy1 raises him above ,pythagoras and ,plato1 and persuades him1 with presumptuous confidence1 to enter the lists against these venerable heroes4 ,and lastly1 all modern languages are barbarous with respect to the ,greek2 falling far short of its harmony and energy1 its copiousness and propriety4 ,if such then be the true state of the case1 what judgment must we form of men who1 with all these disadvantages1 philosophized without the assistance of the ancients1 despising their works1 and being ignorant of their contents8 ,shall we call it prudence or presumption1 wisdom or folly8 ,truth will certainly pronounce the latter2 and the general voice of posterity will confirm her decision4 ,there are two egregious instances in our own country of this daring presumption2 ,i mean ,bacon and ,locke4 ,the former of these is celebrated for having destroyed the jargon of the schoolmen1 and brought experimental enquiries into repute2 and for attempting to investigate causes through the immensity of particular effects4 ,hence1 he fondly expected1 by experiment piled on experiment1 to reach the principle of the universe2 not considering that his undertaking was as ridiculous as that of the giants of old1 who attempted to invade the heavens1 by placing ,ossa upon ,pelion1 and ,olympus upon ,ossa2 and ignorant that ,heaven still1 with laughter1 the vain toil surveys1 ,and buries madmen in the heaps they raise4 ,the latter of these1 ,mr4 ,locke1 is applauded for having1 without assistance from the ancients1 explained the nature1 and exhibited the genuine theory of human understanding4 ,but that this applause is false1 the preceding comparison between his and the ancient philosophy1 may evince2 and the variety of other self-taught systems which1 like nocturnal meteors1 blaze for a while1 and then vanish in obscurity1 abundantly confirms4 ,had these men1 indeed1 when they justly derided the barbarous writings of the schoolmen1 explored the works of antiquity1 penetrated the wisdom they contain1 and enriched their native language with its illustration1 they had doubtless been celebrated by the latest posterity3 but1 desirous of becoming masters in philosophy by intuition1 they disdained the instruction of the ancients1 and vainly attempted to soar on artificial wings to the very summit of science4 ,they are1 however1 destined1 like ,icarus1 to a precipitate fall2 for the influence of time1 which is continually dissolving the cement of their plumes1 is likewise continually weakening their force1 and will at last effect their final separation4 ,and thus much concerning the doctrine of ideas1 and numbers1 according to ,pythagoras and ,plato4 ,,section ,,ii4#bb ,but let us now consider the properties of the demonstrative syllogism1 and endeavour to unravel its intricate web2 appointing ,aristotle for our guide in this arduous investigation4 ,for an enquiry of this kind is naturally connected with the doctrine of ideas1 as it enables us to gain a glimpse of the universals participated in mathematical forms1 and to rise to the principles of science4 ,it brings us acquainted with the laws which bind demonstration2 and teaches us that objects of intellect are alone the objects of science1 and the sources of truth4 ,previous to the acquisition of all learning and ratiocinative discipline1 it is necessary we should possess certain natural principles of knowledge1 as subservient to our future progress and attainments4 ,thus1 in every science there are some things which require an immediate assent as soon as proposed2 whose certainty is too evident and illustrious to stand in need of any demonstrative proof deduced from that particular science which1 like stately pillars1 they equally support and adorn4 ,hence we are informed by the geometrician1 that a point is that which is destitute of all parts whatever2 but we must previously understand the meaning of the word part4 ,thus the arithmetician defines an odd number1 that which is divided according to unequal parts2 but it is necessary we should antecedently know the meaning of the word unequal4 ,thus1 too1 art as well as science operates by antecedent knowledge2 and hence the architect1 the statuary1 and the shipwright1 learn the names and the use of their respective implements1 previous to the exercise of the materials themselves4 ,this is particularly evident in the discursive arts of rhetoric and logic2 thus the logician reasons by syllogism1 the rhetorician by induction1 and the sophist by digressions and examples2 while each proceeds in an orderly progression from principles simple and evident1 to the most remote and complicated conclusions4 #b4 ,the antecedent knowledge of things may be divided into two parts3 the one a knowledge of their existence1 or that they exist2 the other a knowledge of the terms expressive of their existence4 ,thus1 previous to the enquiry why iron is attracted by the magnet1 it is necessary we should learn the reality of this attraction1 and the general mode of its operation3 thus too1 in an enquiry concerning the nature of motion and time1 we must be previously convinced of their existence in the nature of things4 ,the second division of antecedent knowledge takes place in subjects whose very existence admits of a dispute3 thus previous to a solution of the questions1 ,whether there are any gods or not8 ,whether there is a providence or not8 and the like1 it is necessary we should first understand the meaning of the terms2 since we in vain investigate the nature of any thing while we are ignorant of the meaning of its name2 although1 on the contrary1 we may have a perfect conception of the meaning of some words1 and yet be totally ignorant whether the things they express have a real1 or only an imaginary existence4 ,thus1 the meaning of the word centaur is well understood by every one2 but its existence is questioned by most4 #c4 ,from hence it will easily appear1 that no small difference subsists between learning and knowledge4 ,he who is about to understand the truth of any proposition1 may be said to possess a previous conception of its truth2 while1 on the contrary1 it may happen that he who is in the capacity of a learner1 has no antecedent knowledge of the science he is about to learn4 ,thus we attain to the distinct knowledge of a thing which we formerly knew in a general way2 and frequently1 things of which we were ignorant are learned and known in the same instant4 ,of this kind are the things contained under some general idea1 of which we possess a previous knowledge3 thus1 he who already knows that the three interior angles of every triangle are equal to two right1 and is as yet ignorant that some particular figure delineated on paper is a triangle1 is no sooner convinced from inspection of its being a triangle1 than he immediately learns and knows3 he learns it is a triangle2 he knows the equality of its angles to two right ones4 ,that it is now a triangle he both sees and learns2 but the equality of its angles he previously knew in that general and comprehensive idea1 which embraces every particular triangle4 ,indeed1 a definite knowledge of this triangle requires two conditions3 the one1 that it is a triangle2 and the other1 that it has angles equal to two right4 ,the first we receive from inspection2 the second is the result of a syllogistic process2 an operation too refined for the energies of sense1 and alone the province of intellect and demonstration4 ,but demonstration without the knowledge of that which is universal1 cannot subsist2 and since the proposition is universal1 that in every triangle the angles are equal to two right2 as soon as any figure is acknowledged to be a triangle1 it must necessarily possess this general property4 ,hence we infer1 that of the triangle delineated on paper1 and concealed1 we are partly ignorant of this general property1 the equality of its angles 7because we are ignorant of its existence72 and we partly understand it as included in that universal idea we previously possessed4 ,hence too1 it is evident that actual science arises from a medium between absolute ignorance and perfect knowledge2 and that he who possesses the principles of demonstration1 possesses in capacity the conclusions also1 however complicated and remote2 and that by an evocation of these principles from dormant power into energy1 we advance from general and abstracted knowledge to that which is sensible and particular4 #d4 ,two acceptations of knowledge may be admitted2 the one common and without any restriction2 the other limited and peculiar4 ,since all knowledge1 whether arising from accidents1 or supported by necessary principles1 is called science4 ,knowledge1 properly so called1 arises from a possession of that cause from which a thing derives its existence1 and by which we infer the necessity of its existence2 and this constitutes simple and absolute science4 ,thus too1 the definitions of those general conceptions and suppositions1 which from their primary nature are incapable of demonstration1 are called science4 ,but the science which treats of the method of arriving at knowledge1 is called demonstration2 for every demonstration is a syllogism producing science4 ,hence1 if in every syllogism it is necessary that the propositions should be the cause of the conclusion2 and to know any thing properly1 a knowledge of its cause is requisite2 in the propositions of demonstration1 both these conditions are required3 that they should be effective of the conclusion2 and the causes of the thing demonstrated4 ,thus1 from the ruins of a stately edifice1 we may justly infer1 that the building was beautiful when entire2 and from the smoke we may collect the existence of the fire1 though concealed3 but the ruins of the edifice are not the cause of its beauty2 nor does fire originate from smoke1 but1 on the contrary1 smoke is the natural result of fire3 the inference1 therefore1 is in neither case a demonstrative one4 ,again1 since every cause is both prior to1 and more excellent than its effect1 it is necessary that the propositions should be more peculiar1 primary1 and excellent than the conclusions4 ,and because we then know a thing properly when we believe it to have a necessary existence1 hence it is requisite that the propositions should be true2 for if false1 a false conclusion may ensue1 such as1 that the diameter of a square is commensurable with its side4 ,but if every science arises from antecedent knowledge1 demonstration must be founded on something previous2 and on this account it is requisite that the propositions should be more known than the conclusions4 ,the necessary properties1 then1 of all demonstrative propositions1 are these2 that they exist as causes1 are primary1 more excellent1 peculiar1 true1 and known1 than the conclusions4 ,indeed1 every demonstration not only consists of principles prior to others1 but of such as are eminently first2 for if the assumed propositions may be demonstrated by other assumptions1 such propositions may1 indeed1 appear prior to the conclusions1 but are by no means entitled to the appellation of first4 ,but others1 on the contrary1 which require no demonstration1 but are of themselves probable or manifest1 are deservedly esteemed the first1 the truest1 and the best4 ,such indemonstrable truths were called by the ancients1 axioms1 from their majesty and authority2 as the assumptions which constitute the best syllogisms derive all their force and efficacy from these4 ,and on this account1 above all others1 they merit the title of the principles of demonstration4 ,but here it is worth observing1 that these primary propositions are not the first in the order of our conceptions2 but first to nature1 or in the nature of things4 ,to us1 that which is first is particular1 and subject to sensible inspection2 to nature1 that which is universal1 and far remote from the apprehension of sense4 ,demonstration does not submit itself to the measure of our ingenuity1 but1 with invariable rectitude1 tends to truth as its ultimate aim2 and without stopping to consider what our limited powers can attain1 it alone explores and traces out the nature of a thing1 though to us unperceived and unknown4 ,this demonstrative syllogism differs not a little from others1 by the above property2 the rest can as well educe a true conclusion from false premises1 which is frequent among the rhetoricians1 as that which is prior from that which is posterior2 such as1 ,is every syllogism derived from conjecture8 ,with respect to the rest1 as we have already confessed1 they may be formed from principles that are true1 but not from such as are proper and peculiar2 as if a physician should endeavour to prove an orbicular wound the most difficult to coalesce and heal1 because its figure is of all others1 the most capacious2 since the demonstration of this is not the province of the physician1 but of the geometrician alone4 #e4 ,that proposition is called immediate1 which has none superior to itself1 and which no demonstration whatever can confirm3 such as these are held together by the embraces of universals4 ,there are some1 indeed1 united from that which is sensible and particular3 thus1 that the garment is white1 is an immediate proposition1 but not of that kind whose principles require to be demonstrative ones2 the cause of which we shall hereafter investigate4 ,of immediate propositions subservient to the purposes of demonstration1 some are of such a superior nature1 that all men possess a knowledge of them without any previous instruction2 and these are called axioms1 or general notions2 for without these all knowledge and enquiry is vain4 ,another species of immediate propositions is position2 incapable of being strengthened by demonstration1 yet not necessarily foreknown by the learner1 but received from the teacher4 ,with respect to the genus of position1 one of its species is definition1 and another hypothesis4 ,definition is an oration1 in which we neither speak of the existence1 nor non-existence of a thing2 but alone determine its nature and essence4 ,it is common to every hypothesis1 not to be derived from nature1 but to be the entire result of the art of the preceptor4 ,it likewise always affirms the existence or non-existence of its subject3 such as1 that motion is1 and that from nothing nothing is produced4 ,those which are not so perspicuous are called postulates1 or petitions2 as that a circle may be described from any centre1 and with any radius2 and such as these are properly hypotheses and postulates4 #f4 ,we have now seen the privilege assigned to the principles of demonstration3--whether or no our decision has been just1 the ensuing considerations will evince4 ,we said that the assumptions in demonstration were more known than the conclusions1--not indeed without reason1 since through these our knowledge and belief of the conclusion arises4 ,for universally1 that quality which is attributed to many different things so as to be assigned to one through the medium of another1 abounds most in that medium by which it is transmitted to the rest4 ,thus the sun1 through the medium of the moon1 illuminates the earth by night2 thus the father loves the preceptor through the medium of his child4 ,and in the first instance the moon is more lucid than any object it enlightens3 in the second1 the child possesses more of the father's regard than his preceptor4 ,if then we assent to the conclusions through our belief of the principles alone1 it is necessary that the principles should be more known1 and inherit a greater degree of our assent4 ,hence1 if it be true that the principles are more known than the conclusions1 it follows1 that either our knowledge of them is derived from demonstration1 or that it is superior to any demonstrative proof2 and after this manner we must conceive of those general self-evident notions which1 on account of their indemonstrable certainty1 are deservedly placed at the top of all human science4 ,these propositions not only possess greater credibility than their conclusions1 they likewise inherit this property as an accession to their dignity and importance2 that no contrary propositions deserve greater belief2 for if you give no more assent to any principle than to its contrary1 neither can you give more credit to the conclusion deduced from that principle than to its opposite4 ,were this the case1 the doctrine of these propositions would immediately lose its invariable certainty4 #g4 ,there are1 indeed1 some who1 from erroneously applying what we have rightly determined1 endeavour to take away the possibility of demonstration4 ,from the preceding doctrine it appears that the principles are more aptly known than the conclusions4 ,this is not evident to some1 who think nothing can be known by us without a demonstrative process2 and consequently believe that the most simple principles must derive all their credit from the light of demonstration4 ,but if it be necessary that all assumptions should be demonstrated by others1 and these again by others2 either the enquiry must be continued to infinity1 7but infinity can never be absolved71 or if1 wearied by the immense process1 you at length stop1 you must doubtless leave those propositions unknown1 whose demonstration was declined through the fatigue of investigation4 ,but how can science be derived from unknown principles8 ,for he who is ignorant of the principles1 cannot understand the conclusions which flow from these as their proper source1 unless from an hypothesis or supposition of their reality4 ,this argument of the sophists is1 indeed1 so far true1 that he who does not understand that which is first in the order of demonstration1 must remain ignorant of that which is last3--,but in this it fails1 that all knowledge is demonstrative2 since this is an assertion no less ridiculous than to maintain that nothing can be known4 ,for as it is manifest that some things derive their credit and support from others1 it is equally obvious that many1 by their intrinsic excellence1 possess indubitable certainty and truth2 and command our immediate assent as soon as proposed4 ,they inherit1 indeed1 a higher degree of evidence than those we assent to by the confirmation of others2 and these are the first principles of demonstration3 propositions indisputable1 immediate1 and perspicuous by that native lustre they always possess4 ,by means of these1 we advance from proposition to proposition1 and from syllogism to syllogism1 till we arrive at the most complicated and important conclusions4 ,others1 willing to decline this infinite progression1 defend the necessity of a circular or reciprocal demonstration4 ,but this is nothing more than to build error upon error1 in order to attain the truth2 an attempt no less ridiculous than that of the giants of old4 ,for since1 as we shall hereafter accurately prove1 demonstration ought to consist from that which is first1 and most known2 and since it is impossible that the same thing should be to itself both prior and posterior3 hence we infer the absurdity of circular demonstration2 or those syllogisms in which the conclusions are alternately substituted as principles1 and the principles as conclusions4 ,it may1 indeed1 happen1 that the same thing may be both prior and posterior to the same2 but not at one and the same time1 nor according to the same mode of existence4 ,thus1 what is prior in the order of our conceptions1 is posterior in the order of nature2 and what is first in the arrangement of things1 is last in the progressions of human understanding4 ,but demonstration always desires that first which is prior in the order and constitution of nature4 ,but the folly of such a method will more plainly appear from considering its result3 let us suppose every a is b1 and every b is c2 hence we justly infer1 that every a is c4 ,in like manner1 if we prove that every a is b1 and by a circular demonstration1 that every b is a1 the consequence from the preceding is no other than that every a is a2 and thus the conclusion terminates in that from which it first began2 a deduction equally useless and ridiculous4 ,however1 admitting that1 in the first figure1 circular demonstration may be in some cases adopted1 yet this can but seldom happen from the paucity of reciprocal terms4 ,but that reciprocal terms are very few1 is plain from hence3 let any species be assumed1 as man2 whatever is the predicate of man1 is either constitutive of his essence1 or expressive of some accident belonging to his nature4 ,the superior genera and differences compose his essence1 among which no equal predicate can be assigned reciprocable with man1 except the ultimate differences which cannot be otherwise than one1 i4 e41 risibility1 which mutually reciprocates with its subject2 since every man is risible1 and whatever is risible is man4 ,of accidents some are common1 others peculiar2 and the common are far more in number than the peculiar2 consequently the predicates which reciprocate with man1 are much fewer than those which do not reciprocate4 #h4 ,it is now necessary to enumerate the questions pertaining to demonstration2 and for this purpose1 we shall begin with propositions1 since from these1 syllogisms are formed2 and since every proposition consists of a subject and predicate1 the modes of predication must be considered1 and these are three which ,i call total1 essential1 and universal2 a total predication takes place when that which is affirmed or denied of one individual is affirmed or denied of every individual comprehended under the same common species4 ,thus1 animal is predicated of every man1 and it has this farther property besides1 that of whatever subject it is true to affirm man1 it is at the same time true to affirm animal4 ,those things are said to be essentially predicated2 first1 when the predicate is not only total1 but constitutes the essence of the subject2 instances of this kind are1 animal of man2 tree of the plantain2 a line of a triangle2 for a triangle is that which is contained under three right-lines4 ,but here we must observe1 that not every total predicate is an essential one2 thus1 whiteness is predicated of every swan1 because it is inherent in every swan1 and at every instant of time2 but because whiteness does not constitute the essence of a swan1 it is not essentially predicated2 and this1 first1 is one of the modes of essential predication of the greatest importance in demonstration4 ,the second mode is of accidents1 in the definition of which their common subject is applied3 thus1 a line is essentially inherent in rectitude1 because in its geometrical definition1 a line is adopted2 for rectitude is no other than a measure1 equally extended between the points of a line4 ,in the same manner1 imparity is contained in number2 for what is that which is odd1 but a number divided into unequal parts8 ,thus1 virtues are resident in the soul1 because1 in their definition1 either some part of the soul1 or some one of its powers1 is always applied4 ,the third mode of essential predicates pertains to accidents which are inseparably contained in some particular subject1 so as to exclude a prior existence in any other subject2 such as colour in superficies4 ,the fourth mode is of things neither contained in another1 nor predicated of others2 and such are all individuals1 as ,callias1 ,socrates1 ,plato4 ,causes are likewise said to exist substantially1 which operate neither from accident nor fortune4 ,thus1 digging up the ground for the purposes of agriculture1 may be the cause of discovering a treasure1 but it is only an accidental one4 ,but the death of ,socrates1 in despite of vigilance1 is not the result of a fortuitous cause1 but of an essential one1 viz4 the operation of poison4 #i4 ,these posterior significations of essential predicates are added more for the sake of ornament than use2 but the two former have a necessary existence1 since they cannot but exist in the definition of names which predicate the essence of a thing1 and in subjects which are so entirely the support of accidents1 that they are always applied in their definition4 ,but it is a doubt with some1 whether those accidents are necessary1 which cannot be defined independent of their common subject8 ,to this we answer1 that no such accident can1 from its nature1 be contained in every individual of any species2 for curvature is not contained in every line2 nor imparity in every number2 from whence we infer1 that neither is curvature necessarily existent in a line1 nor parity in number4 ,the truth of this is evident from considering these accidents abstracted from their subjects2 for then we shall perceive that a line may exist without curvature1 and number without imparity4 ,again1 ,i call that an universal predicate1 which is predicated of a subject totally and essentially1 and considered as primarily and inseparably inherent in that subject3 for it does not follow that a predicate1 which is total1 should be immediately universal2 for whiteness is affirmed of every swan1 and blackness of every crow1 yet neither universally4 ,in like manner1 a substantial predicate is not consequently an universal one2 for the third mode of essential predicates1 and the two following 7instanced before7 cannot be universal4 ,thus1 colour1 although inherent in superficies essentially1 is not inherent in every superficies1 and consequently not universally4 ,thus again1 ,socrates1 ,callias1 and ,plato1 though they exist essentially1 are not universals1 but particulars2 and thus1 lastly1 the drinking of poison was an essential cause of the death of ,socrates1 but not an universal one1 because ,socrates might have died by other means than poison4 ,if then1 we wish to render an accurate definition of an universal predicate1 we must not only say it is total and essential1 but that it is primarily present to its subject and no other4 ,thus1 the possession of angles equal to two right1 primarily belongs to a triangle2 for this assertion is essentially predicated of triangle1 and is inherent in every triangle4 ,this property1 therefore1 is not universally in figure1 because it is not the property of every figure1 not of a square1 for instance2 nor as universal in a scalene triangle3 for although it is contained in every scalene1 and in every equilateral1 and isosceles triangle1 yet it is not primarily contained in them1 but in triangle itself2 because these several figures inherit this property1 not from the particular species to which they belong1 but from the common genus triangle4 ,and thus much concerning total1 essential1 and universal predicates4 #aj4 ,concerning that which is universal1 we are frequently liable to err2 often from a belief that our demonstration is universal1 when it is only particular2 and frequently from supposing it particular when it is1 on the contrary1 universal4 ,there are three causes of this mistake2 the first1 when we demonstrate any particular property of that which is singular and individual1 as the sun1 the earth1 or the world4 ,for since there is but one sun1 one earth1 and one world1 when we demonstrate that the orb of the earth possesses the middle place1 or that the heavens revolve1 we do not then appear to demonstrate that which is universal4 ,to this we answer3 when we demonstrate an eclipse of the sun to arise from the opposition of the moon1 we do not consider the sun as one particular luminary1 but we deduce this consequence as if many other suns existed besides the present4 ,just as if there were but one species of triangles existed2 for instance1 the isosceles2 the equality of its angles at the base would not be considered in the demonstration of the equality of all its angles to two right ones3 but its triangularity would be essential1 supposing every species of triangles but the isosceles extinct1 and no other the subject of this affection4 ,so when we prove that the sun is greater than the earth1 our proof does not arise from considering it as this particular sun alone1 but as sun in general2 and by applying our reasoning to every sun1 if thousands besides the present should enlighten the world4 ,this will appear still more evident1 if we consider that such conclusions must be universal1 as they are the result of an induction of particulars3 thus1 he who demonstrates that an eclipse of the sun arises from the opposition of the moon between the sun and earth1 must previously collect1 by induction1 that when any luminous body is placed in a right-line with any two others opaque1 the lucid body shall be prevented1 in a greater or less degree1 from enlightening the last of these bodies1 by the intervention of the second2 and by extending this reasoning to the sun and earth1 the syllogism will run thus3 ,every lucid body placed in a right-line with two others opaque1 will be eclipsed in respect of the last by the intervention of the second2 ,the sun1 or every sun1 is a luminous body with these conditions2 ,and consequently the sun1 and so every sun1 will be eclipsed to the earth by the opposition of the moon4 ,hence1 in cases of this kind1 we must ever remember1 that we demonstrate no property of them as singulars1 but as that universal conceived by the abstraction of the mind4 ,another cause of deception arises1 when many different species agree in one ratio or analogy1 yet that in which they agree is nameless4 ,thus number1 magnitude1 and time1 differ by the diversities of species2 but agree in this1 that as any four comparable numbers correspond in their proportions to each other1 so that as the first is to the second1 so is the third to the fourth2 or alternately1 as the first to the third2 so is the second to the fourth3 in a similar manner1 four magnitudes1 or four times1 accord in their mutual analogies and proportions4 ,hence1 alternate proportion may be attributed to lines as they are lines1 to numbers as they are numbers1 and afterwards to times and to bodies1 as the demonstration of these is usually separate and singular2 when the same property might be proved of all these by one comprehensive demonstration1 if the common name of their genus could be obtained3 but since this is wanting1 and the species are different1 we are obliged to consider them separately and apart2 and as we are now speaking of that universal demonstration which is properly one1 as arising from one first subject2 hence none of these obtain an universal demonstration1 because this affection of alternate proportion is not restricted to numbers or lines1 considered in themselves1 but to that common something which is supposed to embrace all these1 and is destitute of a proper name4 ,thus too we may happen to be deceived1 should we attempt to prove the equality of three angles to two right1 separately1 of a scalene1 an isosceles1 and an equilateral triangle1 only with this difference1 that in the latter case the deception is not so easy as in the former2 since here the name triangle1 expressive of their common genus1 is assigned4 ,a third cause of error arises from believing that to demonstrate any property inherent after some particular manner in the whole of a thing1 is to demonstrate that property universally inherent4 ,thus1 geometry proves#bc that if a right-line falling upon two right-lines makes the outward angle with the one line a right-angle1 and the inward and opposite angle with the other a right one1 those two right-lines shall be parallel1 or never meet1 though infinitely extended4 ,this property agrees to all lines which make right-angles3 but they are not primarily equidistant on this account1 since1 if they do not each make a right-angle1 but the two conjointly are equal to two right1 they may still be proved equidistant4 ,this latter demonstration1 then1 is primarily and universally conceived2 the other1 which always supposes the opposite angles right ones1 does not conclude universally2 though it concludes totally of all lines with such conditions3 the one may be said to conclude of a greater all2 the other of a lesser4 ,it is this greater all which the mind embraces when it assents to any self-evident truth2 or to any of the propositions of ,euclid4 ,but by what method may we discover whether our demonstration is of this greater or lesser all8 ,we answer1 that general affection which constitutes universal demonstration is always present to that subject1 which when taken away1 the predicate is immediately destroyed1 because the first of all its inherent properties4 ,thus1 for instance1 some particular sensible triangle possesses these properties3--it consists of brass2 it is scalene2 it is a triangle4 ,the query is1 by which of these we have just now enumerated1 this affection of possessing angles equal to two right is predicated of the triangle8 ,take away the brass1 do you by this means destroy the equality of its angles to two right ones8 ,certainly not3--take away its scalenity1 yet this general affection remains3 lastly1 take away its triangularity1 and then you necessarily destroy the predicate2 for no longer can this property remain1 if it ceases to be a triangle4 ,but perhaps some may object from this reasoning1 such a general affection extends to figure1 superficies1 and extremities1 since1 if any of these are taken away1 the equality of its angles to two right can no longer remain4 ,it is true1 indeed1 that by a separation of figure1 superficies1 and terms1 from a body1 you destroy all the modes and circumstances of its being2 yet not because these are taken away1 but because the triangle1 by the separation of these1 is necessarily destroyed2 for if the triangle could still be preserved without figure1 superficies1 and terms1 though these were taken away it would still retain angles equal to two right2 but this is impossible4 ,and if all these remain1 and triangle is taken away1 this affection no longer remains4 ,hence the possession of this equality of three angles to two right1 is primarily and universally inherent in triangle1 since it is not abolished by the abolition of the rest3----such as to consist of brass2 to be scalene1 or the like4 ,neither does it derive its being from the existence of the rest alone2 as figure1 superficies1 terms2 since it is not every figure which possesses this property1 as is evident in such as are quadrangular1 or multangular4 ,and thus it is preserved by the preservation of triangle1 it is destroyed by its destruction4 #aa4 ,from the principles already established1 it is plain that demonstration must consist of such propositions as are universal and necessary4 ,that they must be universal1 is evident from the preceding2 and that they must be necessary1 we gather probably from hence2 that in the subversion of any demonstration we use no other arguments than the want of necessary existence in the principles4 ,we collect their necessity demonstratively1 thus2 he who does not know a thing by the proper cause of its existence1 cannot possess science of that thing2 but he who collects a necessary conclusion from a medium not necessary1 does not know it by the proper cause of its existence1 and therefore he has no proper science concerning it4 ,thus1 if the necessary conclusion c is a1 be demonstrated by the medium ,b1 not necessary2 such a medium is not the cause of the conclusion2 for since the medium does not exist necessarily1 it may be supposed not to exist2 and at the time when it no longer exists1 the conclusion remains in full force2 because1 since necessary1 it is eternal4 ,but an effect cannot exist without a cause of its existence2 and hence such a medium can never be the cause of such a conclusion4 ,again1 since in all science there are three things1 with whose preservation the duration of knowledge is connected2 and these are1 first1 he who possesses science2 secondly1 the thing known2 and thirdly1 the reason by which it is known2 while these endure1 science can never be blotted from the mind1 but on the contrary1 if science be ever lost1 it is necessary some of these three must be destroyed4 ,if then you infer that the science of a necessary conclusion may be obtained from a medium not necessary1 suppose this medium1 since capable of extinction1 to be destroyed2 then the conclusion1 since necessary1 shall remain2 but will be no longer the object of knowledge1 since it is supposed to be known by that medium which is now extinct4 ,hence1 science is lost1 though none of the preceding three are taken away2 but this is absurd1 and contrary to the principles we have just established4 ,the thing known remains2 for the conclusion1 since necessary1 cannot be destroyed2--he who knows still remains1 since neither dead1 nor forgetful of the conclusion3--lastly1 the demonstration by which it was known1 still survives in the mind2 and hence we collect1 that if science be no more after the corruption of the medium1 neither was it science by that medium before its corruption2 for if science was ever obtained through such a medium1 it could not be lost while these three are preserved4 ,the science1 therefore1 of a necessary conclusion can never be obtained by a medium which is not necessary4 #ab4 ,from hence it is manifest1 that demonstrations cannot emigrate from one genus to another2 or by such a translation be compared with one another4 ,such as1 for instance1 the demonstrations of geometry with those of arithmetic4 ,to be convinced of this1 we must rise a little higher in our speculations1 and attentively consider the properties of demonstration3 one of which is1 that predicate which is always found in the conclusion1 and which affirms or denies the existence of its subject3 another is1 those axioms or first principles by whose universal embrace demonstration is fortified2 and from whose original light it derives all its lustre4 ,the third is the subject genus1 and that nature of which the affections and essential properties are predicated2 such as magnitude and number4 ,in these subjects we must examine when1 and in what manner a transition in demonstrations from genus to genus may be allowed4 ,first1 it is evident1 that when the genera are altogether separate and discordant1 as in arithmetic and geometry1 then the demonstrations of the one cannot be referred to the other4 ,thus1 it is impossible that arithmetical proofs can ever be accommodated with propriety to the accidents of magnitudes3 but when the genera1 as it were1 communicate1 and the one is contained under the other1 then the one may transfer the principles of the other to its own convenience4 ,thus1 optics unites in amicable compact with geometry1 which defines all its suppositions2 such as lines that are right1 angles acute1 lines equilateral1 and the like4 ,the same order may be perceived between arithmetic and music3 thus1 the double1 sesquialter1 and the like1 are transferred from arithmetic1 from which they take their rise1 and are applied to the measures of harmony4 ,thus1 medicine frequently derives its proofs from nature1 because the human body1 with which it is conversant1 is comprehended under natural body4 ,from hence it follows1 that the geometrician cannot1 by any geometrical reasons demonstrate any truth1 abstracted from lines1 superficies1 and solids2 such as1 that of contraries there is the same science2 or that contraries follow each other2 nor yet such as have an existence in lines and superficies1 but not an essential one1 in the sense previously explained4 ,of this kind is the question1 whether a right-line is the most beautiful of lines8 or whether it is more opposed to a line perfectly orbicular1 or to an arch only4 ,for the consideration of beauty1 and the opposition of contraries1 does not belong to geometry1 but is alone the province of metaphysics1 or the first philosophy4 ,but a question here occurs1 ,if it be requisite that the propositions which constitute demonstration should be peculiar to the science they establish1 after what manner are we to admit in demonstration those axioms which are conceived in the most common and general terms2 such as1 if from equal things you take away equals1 the remainders shall be equal3----as likewise1 of every thing that exists1 either affirmation or negation is true8 ,the solution is this3 such principles1 though common1 yet when applied to any particular science for the purposes of demonstration1 must be used with a certain limitation4 ,thus the geometrician applies that general principle1 if from equal things1 c4 not simply1 but with a restriction to magnitudes2 and the arithmetician universally to numbers4 ,thus too1 that other general proposition3----of every thing1 affirmation or negation is true2 is subservient to every art1 but not without accommodation to the particular science it is used by4 ,thus number is or is not1 and so of others4 ,it is not then alone sufficient in demonstration that its propositions are true1 nor that they are immediate1 or such as inherit an evidence more illustrious than the certainty of proof2 but1 besides all these1 it is necessary they should be made peculiar by a limitation of their comprehensive nature to some particular subject4 ,it is on this account that no one esteems the quadrature of ,bryso#bd1 a geometrical demonstration1 since he uses a principle which1 although true1 is entirely common4 ,previous to his demonstration he supposes two squares described1 the one circumscribing the circle1 which will be consequently greater2 the other inscribed1 which will be consequently less than the given circle4 ,hence1 because the circle is a medium between the two given squares1 let a mean square be found between them1 which is easily done from the principles of geometry2 this mean square1 ,bryso affirms1 shall be equal to the given circle4 ,in order to prove this1 he reasons after the following manner3 those things which compared with others without any respect1 are either at the same time greater1 or at the same time less1 are equal among themselves3 the circle and the mean square are1 at the same time1 greater than the internal1 and at the same time less than the external square2 therefore they are equal among themselves4 ,this demonstration can never produce science1 because it is built only on one common principle1 which may with equal propriety be applied to numbers in arithmetic1 and to times in natural science4 ,it is defective1 therefore1 because it assumes no principle peculiar to the nature of the circle alone1 but such a one as is common to quantity in general4 #ac4 ,it is likewise evident1 that if the propositions be universal1 from which the demonstrative syllogism consists1 the conclusion must necessarily be eternal4 ,for necessary propositions are eternal2 but from things necessary and eternal1 necessary and eternal truth must arise4 ,there is no demonstration1 therefore1 of corruptible natures1 nor any science absolutely1 but only by accident2 because it is not founded on that which is universal4 ,for what confirmation can there be of a conclusion1 whose subject is dissoluble1 and whose predicate is neither always1 nor simply1 but only partially inherent8 ,but as there can be no demonstration1 so likewise there can be no definition of corruptible natures2 because definition is either the principle of demonstration1 or demonstration differing in the position of terms1 or it is a certain conclusion of demonstration4 ,it is the beginning of demonstration1 when it is either assumed for an immediate proposition1 or for a term in the proposition2 as if any one should prove that man is risible1 because he is a rational animal4 ,and it alone differs in position from demonstration1 as often as the definition is such as contains the cause of its subjects existence4 ,as the following3 an eclipse of the sun is a concealment of its light1 through the interposition of the moon between that luminary and the earth4 ,for the order of this definition being a little changed1 passes into a demonstration2 thus1 ,the moon is subjected and opposed to the sun3 ,that which is subjected and opposed1 conceals3 ,the moon1 therefore1 being subjected and opposed1 conceals the sun4 ,but that definition is the conclusion of demonstration1 which extends to the material cause2 as in the preceding instance1 the conclusion affirming that the subjection and opposition of the moon conceals the sun1 is a definition of an eclipse including the material cause4 ,again1 we have already proved that all demonstration consists of such principles as are prior in the nature of things2 and from hence we infer1 that it is the business of no science to prove its own principles1 since they can no longer be called principles if they require confirmation from any thing prior to themselves2 for1 admitting this as necessary1 an infinite series of proofs must ensue4 ,on the contrary1 if this be not necessary1 but things most known and evident are admitted1 these must be constituted the principles of science4 ,he who possesses a knowledge of these1 and applies them as mediums of demonstration1 is better skilled in science1 than he who knows only posterior or mediate propositions1 and demonstrates from posterior principles4 ,but here a doubt arises whether the first principles of geometry1 arithmetic1 music1 and of other arts1 can ever be demonstrated8 ,or shall we allow they are capable of proof1 not by that particular science which applies them as principles or causes of its conclusions8 ,if so1 this will be the office of some superior science1--which can be no other than the first philosophy1 to whose charge the task is committed2 and whose universal embrace circumscribes the whole circle of science1 in the same manner as arithmetic comprehends music1 or geometry optics4--,this is no other than that celebrated wisdom which merits the appellation of science in a more simple1 as well as in a more eminent degree than others3 not1 indeed1 that all causes are within its reach1 but such only as are the principal and the best1 because no cause superior to them can ever be found4 ,hence the difficulty of knowing whether we possess science or not1 from the difficulty of understanding whether it is founded on peculiar or common principles2 since it is necessary that both these should be applied in the constitution of all real knowledge and science4 #be;,again1 axioms differ from postulates in this3--they demand our assent without any previous solicitation1 from the illustrious certainty they possess4 ,their truth may1 indeed1 be denied by external speech1 but never from internal connection4 ,he who denies that equal things shall remain from the subtraction of equal1 dissents1 as ,euripides says1 with his tongue1 and not with his heart4 ,but demonstration depends not on external speech1 but on intellectual and internal conviction2 and hence1 axioms derive all their authority from intrinsic approbation1 and not from public proclaim4 ,for the prompt decisions of the tongue are frequently dissonant from the sentiments concealed in the secret recesses of the heart4 ,thus the #bf;geometrician does not speculate those lines which are the objects of corporeal sight1 but such as are exhibited by mental conception1 and of which the delineations on paper1 or in the dust1 are no more than imperfect copies1 notes1 and resemblances4 ,thus1 when he draws a pedal line which is not pedal1 or an equilateral triangle which is not equilateral1 we must pay no regard to the designations of the pen1 but solely attend to the intellection of the mind2 for the property demonstrated of some particular line1 is in the conclusion applied to one that is universal1 and this true line could be no otherwise signified to the learner than by a material description4 ,the certainty of axioms is1 indeed1 in a measure obvious to every one4 ,for what more evident than that nothing exists of which it is possible1 at the same time1 to affirm and deny any circumstance of being8 ,indeed1 so illustrious and indubitable is the light of this axiom1 that in any demonstration we are ashamed to assign it the place of an assumption4 ,it would almost seem prolix and superfluous1 since there is nothing more manifest and certain2 and yet there are cases in which it is necessary to rank it among assumptions4 ,and these take place whenever the intention is to conclude the existence of something as true1 and of its opposite as false4 ,thus1 for instance1 in the demonstration that the world is finite1 we assume this principle1 and then reason as follows3 ,bound and infinite cannot be at the same time affirmed and denied of any body3 ,the world is a body3 ,therefore the world is not at the same time finite and infinite4 ,and in this genus of demonstration1 the major proposition ought always to assimilate with the conclusion4 ,but the above axiom is not the only one obvious1 for the following possesses equal certainty2 that of every thing which exists1 either affirmation or negation is true4 ,this axiom is of great use in demonstrations leading to an absurdity2 for he who demonstrates the impossibility of any opposite assertion1 necessarily establishes his own4 ,hence it is we affirm that the diameter of a square is either commensurable or incommensurable with its side2 and this general principle is accommodated1 and1 as it were1 descends into its proper matter as often as that which it possesses of universal is contracted to a certain genus2 for1 as we have previously observed1 common principles are not admitted in demonstration without any restriction2 but then only when their general nature is limited to some particular subject1 by which they become peculiar and apposite4 #ad4 #bg,wisdom1 or the first philosophy and logic1 agree in not using axioms after the same manner as other arts2 but on the contrary1 they confirm and establish their certainty1 though with this difference1 that the logician reasons only from probabilities1 but the metaphysician from the highest certainty and evidence4 ,besides1 we do not rank logic in the order of the sciences1 because it is destitute of some determinate genus or subject1 as it is neither conversant about lines1 nor numbers1 nor proportions4 ,and its chief concern is about apparent properties1 and not such as are essential to a subject4 ,hence1 in logical disquisitions1 we confidently employ interrogations1 as equally subservient to the affirmation or negation of an opinion3--a method utterly impracticable1 if we only employed those principles which are universally acknowledged2 since it is impossible of the same thing to prove contrary properties1--as of the soul1 that it is mortal and immortal2 but he who demonstrates1 assumes one definite part of a question1 because his purpose is not to interrogate1 but to trace out the latent paths of truth4 ,and hence1 if any one affirms that the soul is moved1 and immediately after denies it1 he is no longer a subject worthy the exercise of our discursive and reasoning powers4 ,again1 it may so happen1 that the same science at one time considers why a thing is1 at another only explains its existence1 or that it exists1 without considering the cause4 ,thus1 the syllogism which concludes by mediate propositions1 demonstrates without assigning the proper cause3 but that which determines by immediate ones1 in a great measure explains the cause or reason of existence4 ,thus1 he who infers that trees do not breathe because they are not animals1 reasons from a mediate and secondary cause1 because there are many animals1 such as insects1 which exist without breathing3 but he who infers this from their want of lungs1 demonstrates from the immediate and primary cause4 ,thus1 the following syllogism is a mediate one1 or such as requires one or more mediums to establish its certainty3 ,every thing that is not an animal does not breathe2 ,a tree is not an animal2 ,therefore a tree does not breathe4 ,here the major proposition is evidently mediate1 because we are still to seek why that which is not an animal does not breathe1 which the following immediate syllogism solves4 ,every thing that is not endued with lungs does not breathe2 ,every thing that is not an animal is not endued with lungs2 ergo1 ,every thing that is not an animal does not breathe4 ,again1 the same science may demonstrate the existence of a thing1 or that it exists1 and the cause of such existence as often as it assigns two immediate reasons2 but the one from the proper cause1 the other only from a sign4 ,thus1 he who demonstrates the increase of the moon1 from the plenitude of her orb1 infers the cause of such increase2 but on the contrary1 he who collects the plenitude of her orb from her increase1 reasons only from a sign1 and can alone declare its existence4 ,and1 indeed1 it often happens that the cause and sign reciprocate1 so that as from the sign we advance to the cause1 demonstration from the cause frequently recurs to the sign4 ,thus1 from the breadth and firmness of the basis1 we collect the permanent duration of the pyramid2 and from its extended existence we infer the strength of its support4 ,whenever1 then1 the argument originates from a sign1 it gives evidence to the conclusion1 as from something more known than its cause4 ,when it begins from the cause1 it proceeds from that which is first in the order of nature1 to that which is last1 and reasons as from the proper principle of the thing4 ,sometimes the cause and sign do not reciprocate4 ,thus1 although wherever there is smoke1 we infer the existence of fire2 yet we cannot infer1 that wherever there is fire smoke exists4 ,thus1 from the palace and the picture we collect the existence of the architect and painter2 but the last may exist without the first2--the living architect without the actual palace2 and the living painter without the energies of his art4 ,and thus it is that the cause is illustrated by its sign2 but not always the sign by its cause4 ,hence then1 as all causes do not reciprocate with their effects2 so neither is it always causes and effects which do reciprocate3 because a multitude of signs1 mutually inferring each other1 may accompany a certain cause4 ,thus1 the signs which attend the causes of a fever1 are a quick pulsation of the artery1 and an intense heat3 and these signs mutually assert each other2 but no syllogism can be composed from either expressing the why1 but only simply that the other exists4 #ae4 ,we now propose to consider the mode in which the two preceding demonstrations are distributed in different sciences4 ,when sciences then are so related1 that the one is dependent on the other1 as optics on geometry1 navigation on astronomy1 and music composed by the arbitration of the ear1 on that which consists in the knowledge of mathematical proportions3 in this case1 the demonstration of simple existence1 or that they exist1 pertains to the science of sensibles2 but the demonstration why they exist to the science which is speculative and mathematical4 ,thus the mathematician speculates the causes of a certain sensible effect1 without considering its actual existence2 for the contemplation of universals excludes the knowledge of particulars2 and he whose intellectual eye is fixed on that which is general and comprehensive1 will think but little of that which is sensible and singular4 ,thus1 by mathematics we may learn the responsive harmony of the last chord1 and its consonance with the mean2 but we cannot perceive this concord1 if unaccustomed to the practice of the musical art4 ,in fine1 those sciences which are more of a mathematical nature1 ,i mean such as are more amply conversant with the inspection of things1 considering their forms abstracted from every material subject1 always demonstrate the why2 and such is geometry in respect of optics4 ,thus geometry considers only such things as are peculiar to right-lines1 independent of every sensible connection4 ,for the geometrician does not investigate a right-line as contained in stone or brass2 but considers it as entirely detached and unconnected with any object of sense4 ,on the contrary1 optics receives a right-line just as it is perceived in a rule1 or engraved in brass4 ,and1 indeed1 in treating of some particulars1 natural science has the same relation to optics1 as optics to geometry4 ,thus1 in considering the reason of the appearance of the rainbow1 the natural philosopher defines the bow to be an image refracted from a certain cloud against the sun2 but why it is endued with such a form1 and seen with such a colour1 must be assigned by him who is skilled in optics4 ,there are1 again1 sciences1 one of which is not subordinate to the other1 because founded on principles totally different2 yet1 in some particulars they agree with the preceding4 ,thus1 to know that an orbicular wound is the most difficult of cure1 belongs to the physician2 but to know why1 to the geometrician4 #af4 ,of all syllogistic figures1 the first is the best adapted to science1 since the arithmetician1 geometrician1 and lastly all those who demonstrate any effect from its proper cause1 fabricate their reasonings according to this figure4 ,for the middle figure is seldom used1 because only adapted to a few occasions3 and since the knowledge of the why is of all others the most important1 which is alone obtained by this figure3 hence1 in the pursuit of science1 it is always preferred before the rest4 ,besides1 it is equally accommodated to the knowledge of final causes2 to which it alone tends3 for it composes definitions from words universal1 and affirmative4 ,in the second figure1 a complex negative is conceived2 and in the last1 a particular one4 ,add to this1 that mediate propositions are no other ways reducible to immediate ones than by this figure1 in which the mediate proposition tends1 by a continued series1 to that which is immediate4 ,but the second does not conclude affirmatively1 nor the last universally2 from whence it appears1 that a mediate proposition can never become immediate by these figures3 not that all affirmative propositions are immediate ones1 since some negatives are of this kind2 for all propositions are equally immediate1 which cannot be confirmed by syllogism2 and such are those negatives1 of whose terms it is impossible any genus can be affirmed4 ,thus the proposition1 no substance is quality1 is an immediate negative of this kind1 whose terms are two of the most universal genera of things4 ,again1 as we have frequently affirmed that he who demonstrates1 always assumes such things as are essentially predicated2 but that he who argues dialectically or topically1 not always1 but generally assumes such as are accidentally predicated1 and which appear more probable and known than such as are essentially inherent2 it is proper we should define what is meant by accidental predication2 or something predicated by means of another4 ,indeed1 the term has a diffuse signification3 for1 first1 a body is said to be white by something else1 because by its superficies2 and in this manner vines are white1 because their branches are white4 ,thus1 if accident be predicated of accident1 it is by means of another2 as when we say the musician is fair2 for the being a musician is an accident of man1 and the being fair of the musician3 and man is the subject of each4 ,the predicate of substance is equally accidental1 when not included in the number of things substantially inherent2 as when we affirm of any particular man that he is red1 or black4 ,but the predication is especially accidental1 as often as1 by perverting the order of nature1 substance is predicated of accident2 as when we say something white is an animal3 for this assertion differs from that other1 animal is white4 ,in the latter1 the subject animal is neither inherent in another1 nor subsists by another1 but has an essential existence4 ,in the former1 what is assumed as a subject derives its existence from that of which it is the accident4 ,it is only dialectically1 therefore1 that we can argue from predicates as probable and known without any distinction3 but in demonstration1 all that are preposterous and accidental must be carefully avoided1 excepting such accidents as being essentially in a subject1 admit of an essential predication2 and some of these we have enumerated before4 #ag4 ,we are now entering on a disquisition neither ignoble nor useless3 it is this1 whether the number of things predicated essentially of a subject is finite1 or whether things in a continued series run on to infinity4 ,for instance1 let us suppose some ultimate subject1 which is not the predicate of any thing besides2 and let c represent such a subject1 of which b is the first and immediate predicate2 and in the same manner d of b1 and e of d3 the query is1 ,whether or not this extraction must necessarily stop1 or will admit of an immense progression1 so that f may be predicated of e1 and g of f1 and so on infinitely2 the power of the predicates1 which supplies the common identity1 still remaining inexhaustible and undiminished8 ,the second query is this1 ,supposing some general subject1 which we call a1 of such a nature as to be no longer the subject of any farther predication1 but to be itself the supreme and primary predicate2 and supposing that it is immediately inherent in f1 and f in e1 and e in g1 whether or not the process must stop1 or extend to infinity1 and no subject be found which is not directly predicable of another8 ,there is a remarkable difference in the two considerations2 for1 in the former we enquire whether any ultimate subject can supply an infinite ascent of predicates2 in the latter1 whether any first predicate can exist in an infinite descending series of subjects4 ,the third question is1 supposing two extremes constituted from a first predicate and last subject1 whether it is possible an infinite number of mediums can intervene8 ,and this is no other than to enquire whether demonstrations admit of an infinite progression1 so that whatever is assumed in proof of another1 must be proved itself8 ,or whether it is not more agreeable to truth1 that there should be some immediate propositions and ultimate terms1 whose discovery may give respite to enquiry1 and stay the elaborate process of demonstration8 ,the same question occurs in negatives4 ,but that some of these are immediate1 the instance lately alledged sufficiently evinces4 ,the solution of this enquiry is not so difficult in subjects which mutually reciprocate2 for in these1 when the ultimate subject is given1 no one can doubt the existence of their primary predicate2 nor when the primary predicate is admitted1 can there be any doubt of the existence of some ultimate subject4 ,for1 in things which mutually reciprocate1 whatever is enquired of the one1 is immediately questioned of the other2 and wherever there is a last subject1 there must be a first predicate2 for by the conversion of the ultimate subject you effect the primary predicate4 ,previous to the discussion of the first question1 it is necessary to know that infinite intermediates cannot intervene between two finite terms in an ascending and descending series of predications4 ,i call the series ascending which rises to universals2 but descending1 which1 by a contrary process1 stops at particulars4 ,thus1 if any one admits that a is some first predicate1 and g some ultimate subject1 and should contend1 that between these terms there may be infinite mediums1 he contradicts himself2 since he who begins from a in a descending progression1 will never1 by this means1 arrive at g2 and he who departs from g in an ascending series1 can never finally rise to a4 ,so that the extremes can be no longer finite1 as the hypothesis admitted4 ,indeed1 the absurdity of such a supposition is the same as to contend that between one and ten1 an infinity of numbers may exist2 which is evidently impossible1 because the discrete nature of numbers excludes their actual existence in infinitum1 between any finite limits2 since they can only become infinite from their actual existence and precedence1 and not from any dormant power or capacity they possess3 for between any two given numbers there is nothing similar to number in capacity1 which can ever become number in energy2 as in quantity continuous between any two points there are always parts in capacity1 which1 whenever a proper agent is at hand1 become immediately actual4 ,in like manner1 he who admits the terms finite1 but believes that the mediums are infinite1 asserts what is impossible1 since these logical predications are of the same discrete nature with numbers themselves4 ,thus all the predicates which can exist between ,socrates and substance1 must exist actually1 or not at all2 for surely between these two terms1 or periods1 no predicate in capacity can ever be supposed to subsist4 ,if it be urged1 that the capacity of receiving these predicates exists between ,socrates and substance1 still we reply1 it is not that kind of capacity in which these predicates can retain the most shadowy existence2 out of which they can ever be called forth into energy1 as from some latent retreat2 or into which they can finally retire1 when energy is no more4 ,and hence we conclude it impossible that infinite mediums can exist between any finite terms4 #ah4 ,it now remains that we prove1 first1 by probable arguments1 and then by such as are demonstrative1 that the extremes in any series of predications are finite2 and that an infinite progression is impossible1 not only in substantial predicates1 but in such as are accidental4 ,for every thing predicated of another is either essentially or accidentally inherent2 and is predicated in a natural or preposterous order4 ,it is predicated according to nature1 when accident is declared of substance2 contrary to nature1 when substance of accident4 ,that essential predicates are finite1 appears from hence1 because a contrary hypothesis excludes the existence of definition1 by admitting that all things are contained in some superior genus1 and acknowledge some farther definition2 since it is impossible that the definitions of genus can ever be circumscribed1 while there is a continual supply of other genera1 which can never be known without definition2 for thus we shall never obtain either a beginning or an end4 ,but to define all things is not possible1 because infinity can never be absolved by the most unwearied progression4 ,predictions then1 of this kind1 are always circumscribed by a certain number of terms1 which prevent their infinite process1 and cause all the strength of demonstration1 and all the certainty of human knowledge4 ,the same may be proved in accidents2 for such as are predicated of substance1 are either predicated as qualities or quantities1 as relatives1 or as actions and passions2 as expressive of some habit1 or significant of some place2 or as connected with some time4 ,thus we say the wood is white1 the triangle is scalene2 whiteness being accidental to the wood1 and scalenity to the triangle4 ,it is therefore certain1 that every accident is predicated of substance2 and it is no less certain that the predicates of substance are finite1 since they are all included in the ten universal genera of things4 #ai4 ,we have hitherto defended the impossibility of an infinite progression of logical predicates and subjects1 in a demonstrative process1 by such arguments as are dialectical and common3 it now remains that we adopt such as are peculiar and certain4 ,demonstrations1 then1 are derived from affections essentially inherent in a subject2 and these are either such as take place in definitions of a subject1 as multitude and quantity1 are essentially predicated of number2 or1 secondly1 accidents which are defined from their subjects1 as imparity by number4 ,but the predication cannot1 in either case1 be extended to infinity4 ,for it is not necessary that in the same manner that imparity is predicated of number1 something else1 suppose c1 should be predicated of imparity2 and so imparity be contained in its definition1 similar to number in the definition of imparity4 ,for in predications of this kind1 the terms are always assumed more contracted than their subject2 and at length1 by a continued procession1 must terminate in an indivisible4 ,thus1 as imparity is more contracted than number1 c must be more contracted than imparity4 ,hence1 these predications either finally stop1 for the reasons we have assigned2 or because whatever is predicated of imparity1 is necessarily predicated of number2 so that one thing as number would be actually contained in the definition of an infinity of things2 and so actual infinity must ensue1 which is absurd4 ,lastly1 whatever is said to reside in the terms1 must be allowed to reside in the subject2 so number must be applied in the definition of every affection2 and an infinite number of properties will be essentially inherent in number2 and number will inherit infinite definitions4 ,but affections essentially resident in a subject cannot be infinite1 because it is necessary they should exist in energy4 ,thus1 imparity cannot exist potentially in number2 nor reason in man2 nor rotundity in a circle1 because wherever these subjects have an actual being1 it is necessary these essential attributes should be actually inherent4 ,again1 in the definitions of a subject1 an infinite process is impossible1 because from such an hypothesis nothing could ever be defined2 and thus it appears that neither can demonstrations be infinitely extended1 nor every thing admit of demonstration1 an opinion we have already noticed in the beginning of this section3 for if neither universally1 nor in every proposition a middle term can be assumed1 but as soon as we arrive at immediate propositions1 the labour of investigation is finished1 the possibility of demonstrating every thing can no longer be defended2 since it is proved above1 that by limiting the extremes1 an infinite number of mediums is necessarily excluded4 ,and thus1 by taking away infinity from the reasoning art1 we have given a support to science1 which the most vigorous efforts of subtle sophistry can never finally subvert4 ,we have set bounds to that restless spirit of enquiry which wanders uncontrouled in the mind unenlightened by science1 by every where circumscribing its progress within the limits of that which is most particular1 and most universal1 a first predicate1 and an ultimate subject3 and finally1 by asserting that all the evidence of human knowledge results from the lustre of primary and immediate principles1 we have held up a steady and permanent light1 ever sufficient to direct our steps through the dark mazes of ignorance and error1 into the bright paths of certainty and truth4 #bj4 ,let us next consider whether universal demonstration is preferable to particular1 or not4 ,and first1 in favour of particulars we may say that their evidence is more exquisite and certain than that of universals4 ,thus1 the knowledge1 from inspection1 that ,callias is a rational animal1 is superior to that acquired by a reasoning process which infers his rationality1 because every man is a rational animal4 ,by particular demonstration a thing is known as it is1 by universal only in common4 ,besides1 particulars possess some solidity1 universals none3 and the demonstration of things which have a real existence1 is more excellent than that of things which have none4 ,and there are no errors more frequent than those about universals2 demonstration considering them as things entirely abstracted from singulars4 ,on the contrary1 particulars are usurped by the sight1 grasped1 as it were1 by the hand1 and the general subject of every sense2 so that concerning these1 demonstration affirms nothing false or inconstant4 ,but these reasons1 however plausible1 are easily confused4 ,and1 first1 the term essential is more closely connected with universals than particulars4 ,thus the possession of three angles equal to two right1 is an affection more essential to the triangle itself1 than to one equilateral or scalene4 ,add too1 that in the demonstration of universals we always infer some property of a subject from its simple existence1 or because it is such a subject4 ,again1 many affections are contained in singulars assumed from no particular nature1 but from that which is universal2 as rationality in ,socrates1 which is not inferred from his existence as ,socrates1 but from his existence as man4 ,farther1 that demonstration is the more excellent which is derived from the better cause3 but an universal cause is more extended and excellent than a particular one2 since the arduous investigation of the why in any subject is stopt by the arrival at universals4 ,thus1 if we desire to know why the exterior angles of a triangle are equal to four right ones1 and it is answered1 because the triangle is isosceles2 we again ask1 ,but why because isosceles8 ,and if it be replied1 because it is a triangle1 we may again enquire1 ,but why because a triangle8 ,to which we finally answer1 because a triangle is a right-lined figure2 and here our enquiry rests at that universal idea which embraces every preceding particular one1 and is contained in no other more general and comprehensive than itself4 ,add too1 that the demonstration of particulars is almost the demonstration of infinites2 of universals1 the demonstration of finites4--,we add farther1 that demonstration is the best1 which furnishes the mind with the most ample knowledge2 and this is alone the province of universals4 ,again1 the principles of science become immediate only in proportion as the demonstration becomes universal2 and he who knows universals1 knows particulars in capacity3 but we cannot infer1 that he who has the best knowledge of particulars1 knows any thing of universals4 ,lastly1 that which is universal1 is the province of intellect and reason1 particulars are the offspring of sense2 and hence we conclude that universal demonstration exceeds particular both in dignity and excellence1 and is first in the nature of things1 although last in the progressions of the reasoning power4 ,again1 ,that affirmative demonstration is superior to negative1 appears from hence3 the affirmative does not require the assistance of the negative2 but the negative cannot exist without the affirmative2 on which account1 the demonstration composed from negatives alone1 is incapable of producing real evidence and conviction4 ,besides1 affirmation exceeds negation both in priority and simplicity of existence4 ,again1 the demonstration which concludes directly1 is better than that which confirms a proposition by evincing the absurdity of its contrary4 ,the first proceeding in a regular order1 establishes1 by a natural deduction1 the truth which was first advanced4 ,the second taking a wider circuit1 yet with the same intentions produces a conclusion quite opposite to its apparent design4 ,the one may be compared to the open attack of a valiant and skilful soldier1 who expects the conquest of his enemy from strength and courage alone3 the progress of the other resembles the same soldier1 uniting force with stratagem1 and advancing1 by an irregular march1 which his foe mistakes for a retreat1 but finds the secret cause of his destruction4 ,the first is simple and impromiscuous1 as composed from propositions alone3 the second is compound and miscellaneous1 calling in hypothesis to its assistance4 #ba4 ,one science is said to be prior to1 and more certain than another in many respects2--when the one reasons from primary causes1 but the other from such as are secondary3--when the one may be ranked in the genera of intelligibles and universals2 but the other in the genera of sensibles and particulars4 ,and such is the relation of arithmetic to music2 of geometry to optics2 and lastly1 of every superior to every subordinate science4 ,again1 this happens when the one reasons from simple principles1 the other from such as are complex and connected2 on which account arithmetic seems to possess greater certainty than geometry4 ,for the principle of arithmetic is unity2 but of geometry a point2 and unity is without position1 with which a point is always connected4 ,and in this manner geometry inherits greater evidence than astronomy2 for the one considers body simply1 the other as connected with a circular motion4 ,the science is called one which contemplates actions belonging to one genus3 the genus is one which possesses the same first principles2 and hence geometry and stereometry form one science4 ,on the contrary1 the sciences are called different which have different principles1 such as geometry and optics2 the latter of which does not originate from the principles of the former4 ,again1 the same thing may admit of many demonstrations1 and may be known from many mediums3 at one time from the application of such as are congenial3 at another1 from those of a different order or genus4 ,from congenials1 as when we demonstrate that the plantain is a substance1 first1 by the medium of a tree1 and then by the medium of a plant1 thus3 ,every tree is a substance2 ,the plantain is a tree3 ,therefore the plantain is a substance4 ,and again1 ,every plant is a substance3 ,the plantain is a plant3 ,therefore the plantain is a substance4 ,we demonstrate1 from mediums1 of a following order or genus1 as when we prove man to be a substance1 at one one time from his being rational1 at another from his being a biped2 and these mediums1 in part1 mutually contain each other4 #bb4 ,fortuitous events can never1 in any science1 become the subject of demonstration2 since they are neither limited by necessity1 nor admit the arrangement of syllogism4 ,indeed1 so far from obtaining a necessary1 they do not possess a frequent existence1 but every syllogism is composed from one or other of these4 ,again1 science is not the business of sense1 since that which is universal is the object of perception in particulars themselves4 ,for the object of sight is colour in general1 and not this particular colour3 the object of hearing is sound in general1 and not any particular sound2 and1 on this account we see or hear not only this or that colour or sound1 but likewise every other which falls under the cognizance of these senses4 ,hence1 if it were possible for any one to discern by his sight1 the equality of the three angles of some particular triangle to two right1 he would not by this means possess a demonstration of the conclusion which affirms this to be the property of every triangle2 but his knowledge would extend no farther than the triangle he inspects4 ,thus too1 if we could perceive an eclipse of the moon to arise from the interposition of the earth1 we could not universally conclude that this is the cause of every eclipse1 but only of the particular one we behold4 ,for the explication of causes extends to universals2 and comprehends not only the knowledge of one particular defect of the moon1 but simply of every eclipse2 since the interposition of the earth is not so much the cause of any present eclipse1 as of all which can possibly exist in every age4 ,whenever1 then1 the cause is universal1 the knowledge of any effect deduced from such a cause is1 in every respect1 superior to the evidence arising from the perceptions of sense4 ,it is likewise more excellent than the apprehension which subsists independent of the proper cause2 as if any one should give absolute credit to the proposition1 that the three angles of a triangle are equal to two right1 without a previous conviction that the external angle of a triangle1 is equal to the two interior opposite ones2 and without applying this last proposition as the cause of the first4 ,the comprehension1 then1 which is conjoined with the proper cause1 far exceeds the strongest evidence of sense4 ,but perhaps it may be said that science consists in sense1 because the science of any particular1 fails from a defect of the sense by which it is apprehended4 ,to this we reply1 that science1 indeed1 is not acquired without the assistance of sense1 but it does not follow from hence1 that to perceive is to know2 because the object of science is that which is universal2 but of sense1 that which is particular4 ,thus1 if we could see light penetrating the pores of glass 7on the atomical hypothesis7 the cause why it illuminates would be manifest from sensible inspection as the means1 and from the universal apprehension of science1 by which we should understand this to be universally true4 ,again1 the principles of all sciences cannot be the same neither considered as remote or proximate4 ,not considered as proximate1 because the principles always correspond to the demonstrated conclusions2 but these are not the same1 since they are often generically different2 and consequently the propositions from which they result must be derived from discordant genera4 ,but propositions consist of such things as essentially exist2 and hence we infer1 that the principles of geometry are essentially distinguished from those of arithmetic1 that they cannot admit of reciprocal accommodation1 so that the one may be predicated1 or become the subject of the other1 and that the one can never be subservient as a medium to the other4 ,again1 common and first principles are not applied in every science2 such as this1 that every thing must either be affirmed or denied4 ,nor can any thing be proved by their assistance alone1 but as often as these are required in demonstration1 other principles more proximate and peculiar to the given proposition1 must always be adopted4 ,again1 axioms universally conceived1 cannot be assumed in syllogism1 but they must be contracted1 as it were1 to some subject genus4 ,of this kind is that common axiom1 that as often as any four quantities are proportionable1 by permutation1 or changing the order of the terms1 the same ratio will result4 ,for the arts apply this axiom in a restricted sense2 geometry1 by considering the relatives as four magnitudes1 and arithmetic as four numbers2 but the natural philosopher1 by adapting the comparison to four motions1 or four times4 ,besides1 if the principles of all sciences were the same1 it is necessary they should be comprehended by some certain number1 similar to the limitation of the elements3 but every science is capable of immense increase from the many different modes of amplification the conclusions will admit2 and consequently it is requisite to establish a correspondent number of proper principles2 for such as are common cannot be alone sufficient4 ,lastly1 if the same principles accord with every science1 it follows1 that any thing may be demonstrated from such principles3 but the certainty of geometrical conclusions cannot be established from the principles of music2 and from hence it follows1 that although the principles of every science are not the same1 they do not possess an entire diversity1 nor yet an absolute affinity of nature4 #bc4 ,there is a remarkable difference between science and opinion4 ,whatever is the subject of science must have a necessary existence2 on the contrary1 opinion is conversant with things liable to mutation and decay4 ,again1 as science depends on necessary propositions for support2 so opinion on such as possess only a possibility of existence2 and so there is one mode of approbation in subjects of opinion1 and another in those of science4 ,hence science is distinguished from opinion by two discriminations1 the one arising from their subjects1 the other from the mode of approbation4 ,that opinion is conversant with things possible or contingent1 we may learn from hence2 contingencies cannot belong to science1 because their existence is not necessary2 nor to intellect1 or that principle of science by which its terms are known2 nor to the apprehension or belief of immediate propositions1 called indemonstrable science4 ,hence1 if every habit by which truth is known1 is either science1 or intellect1 or opinion1 it remains that opinion alone consists of things which are1 indeed1 true2 but not necessary4 ,it is1 therefore1 inconstant and unstable1 from the mutable nature of its subjects4 ,besides1 no one thinks he possesses an opinion of things which he believes to have a necessary existence1 so that they cannot be otherwise than they are2 but to such conviction he properly gives the name of knowledge1 and to its contrary the name of opinion4 ,again1 the same thing from the same propositions may at one time become the subject of knowledge1 at another1 of opinion2 and this happens according to the different formation of the syllogism which the propositions compose2 whether reasoning from the proper cause it explains the why1 or only simply declares a thing exists4 ,hence a doubt arises why opinions of this kind may not be called science1 since both the subjects and propositions are the same8 ,the solution is obvious4 ,if it is believed that the propositions cannot be otherwise than they are1 or that they have a necessary existence1 such an assent of the mind is not opinion1 but science2 because things which inherit an essential existence are the ornaments of science alone4 ,on the contrary1 if we are convinced that the propositions are true1 but at the same time not necessary1 such conviction is not science1 but opinion4 ,hence1 it is impossible that science and opinion can be the same1 since they vary in their definition and mode of approbation1 and in a different manner demand our belief4 ,similar to this1 although it may happen that of the same thing a true and a false opinion may arise1 it will not therefore follow1 that true and false opinions are the same4 ,for that which is firm and constant can never be the same with that which is mutable and frail2 and that which is always true must be essentially different from that which may be changed into false4 ,by the power of habit indeed in different men1 the same thing may be comprehended by opinion and science4 ,thus it was opinion in ,epicurus when he said that the sun was eclipsed by the moon passing under its orb1 because he thought it might otherwise happen1 and that the moon might be interposed without obscuring the light of the sun4 ,it was science in ,hipparchus1 because he knew it as a necessary event4 ,but in the same mind1 at the same time1 and of the same thing1 it is impossible that science and opinion can exist4 ,and thus much concerning the difference of the two4 #bd4 ,lastly1 sagacity is an acute and sudden apprehension of the medium1 or proper cause of a certain effect3 as if any one1 beholding the moon1 should in a moment conjecture the cause of the part opposite to the sun being lucid1 and the other parts obscure1 because she derives her splendor from the sun4 ,hence he is universally called acute and sagacious1 who1 from the aspect or hearing of the extremes1 can readily perceive the medium which exists between them3 as the term imports a certain revolution of the conclusion into its first propositions1 and1 as it were1 a swift comprehension and continuation of the medium4 ,,section ,,iii4 ,in the ensuing ,commentaries1 the soul is considered as immaterial2 and as possessing a middle nature between intelligibles and sensibles2 but that this important assertion may not remain without proof1 the following demonstrative arguments1 derived from the ,platonic philosophy1 are offered to the reader's consideration4 ,and first1 that the soul is an unextended1 and consequently immaterial essence1 may be thus proved4 ,if1 after the manner of magnitude1 it consisted of continuous parts1 it would be impossible that any one part could be sentient from the passion of another2 but the soul1 for instance1 which is situated in the finger1 would be sensible of passion1 as if detached from soul in the other parts of the body1 and existing by itself2 for the soul1 from this hypothesis1 would be sentient by parts1 and not considered as a whole4 ,besides1 there must be many souls governing each part of us1 different among themselves1 and endued with their own peculiar energies4 ,for whatever may be said of continuity1 is to no purpose1 unless it conduces to unity of sensation2 so that the hypothesis which supposes that the sensations gradually arrive at the principal part of the soul by a certain continued succession1 is not to be admitted1 since it may be reasonably asked1 ,how is the principal part to be peculiarly distinguished8 ,by what rule of quantity can the parts be discerned1 by what difference are they to be distinguished1 where the quantity is one1 and the bulk continuous8 ,besides1 is the principal part alone1 or are the other parts1 sentient8 ,if the principal part alone be perceptive1 the soul can then alone be sentient when a sensible passion meets with this principal part1 situated in its peculiar seat2 but if a sensible passion falls upon any other part of the soul destitute of sense1 it is impossible that such a part should be able to transmit the same passion to the principal1 or be at all sentient4 ,for how can that which is void of sense1 receive passion1 and convey it entire to a sentient part8 ,besides1 if passion accedes to the principal1 it either falls on one of its parts1 and so either one part alone will be sentient1 and the rest without sensation1 and consequently superfluous1 or there must be innumerable and dissimilar sensations2 for if the sensation of each of the parts singly1 is the same with the aggregate of them all1 of what use is a multitude of parts8 ,but if the sensations are various1 a man may say1 as it were1 ,i am primarily sentient in this place1 and secondarily in another2 and every sentient part besides the first1 will be ignorant where sensation is most powerful3 or perhaps 7from such an hypothesis7 every part of the soul will be equally deceived1 each part thinking the passion to arise in the place where it is situated4 ,but if not the principal part alone1 but every part of the soul be sentient1 a principal part is superfluous for the purpose of sensation2 and if the soul be divisible like magnitude1 how is it able to recognize1 as belonging to one subject1 the qualities which flow1 as it were1 through many senses1 as through the eyes and ears8 ,for that part of the soul which is perceptive through the eyes1 distinguishes nothing but colours2 that which energises through the ears1 nothing but sounds2 and that which acts through the medium of the touch1 nothing but the surfaces of bodies3 what is it then which perceives all these properties of bodies united in one subject1 or what is that which perceives any thing as a certain one8 ,for unless the intentions of the senses1 and of sensible objects1 were collected together in one1 the soul could never be able to judge of the peculiar and different properties of bodies3 ,hence it is necessary that the soul should be1 as it were1 a general centre2 that the several senses should be extended on all sides to this1 similar to lines verging from the circumference of a circle to the centre2 and that a power of this kind1 comprehending all things2 should be truly one4 ,for if the soul was any thing divisible1 and the intentions of the senses reached the soul1 and ended in its essence like the extremities of various lines1 they must either again concur in one and the same as a medium1 or have different situations1 in such a manner that each sense may perceive different from one another3 as if1 for instance1 the sense of sight should perceive the form of ,socrates1 and the sense of hearing recognize his voice2 that essence which pronounces the whole to be one person1 that of ,socrates1 must be something different from each of the senses4 ,hence it is necessary that the soul should be an indivisible essence2 for if she possessed magnitude1 she must be divided along with every sensible object she perceives2 so that one part of the soul would perceive a certain part of a sensible object1 and we should possess no sentient power capable of perceiving the whole1 or of pronouncing any thing one4 ,thus1 in the perception of a man1 considered as one1 how is it possible that the soul can be divided1 so that the perception of a part shall be the same with the perception of the whole4 ,but if we suppose the soul divisible in all her perceptions1 since it is impossible she should be co-extended with every sentient object1 in how many parts is the division to be made8 ,is the soul to be distributed into the same number of parts as the sensible object she perceives1 so that every part of the soul may perceive the same part of the object8 ,or shall we say that the parts of the soul have no sensation of the parts of the object8 ,but this would be absurd and impossible4 ,if every part then of the soul perceived every part of a magnitude as a whole1 since magnitude is divisible to infinity1 and since1 on this hypothesis1 there must be innumerable sensations of every sensible object1 there must be innumerable images1 as it were1 of the same thing in our principal part4 ,besides1 if that which perceives is corporeal1 it will not be possible for it to perceive in any other manner than as if certain images were impressed from a seal in wax1 or in brass1 or in any other sensible substance4 ,but if the images of sensation exist as in humid bodies 7which is most probable71 they will certainly be confounded like images in water2 nor can there be any memory1 the image departing with its forming substance4 ,and if we suppose the figures to remain like impressions in solid bodies1 either it will not be possible for others to succeed while the former endure2 and thus1 sensations of other things cannot take place2 or if others succeed1 the former images must be immediately destroyed1 and memory be no more4 ,so that if we allow it possible to remember1 and1 besides this1 to perceive other things1 without any hindrance from former impressions1 it is impossible that the soul should be corporeal4 ,since the soul1 therefore1 is an unextended1 indivisible1 and immaterial substance1 it is consequently incorruptible and immortal2 for every thing capable of dissolution and dispersion is either corporeal and composite1 or exists in some subject from which it is inseparable4 ,and indeed1 whatever may be dissolved is corruptible1 as being compounded from many4 ,but whatever naturally subsists in something different from itself1 when it is separated from its subject1 immediately vanishes into non-entity4 ,but the soul1 as we have proved1 is incorporeal2 it is likewise removed from every subject1 and naturally reverts to itself1 and is therefore immortal and incorruptible4 #b4 ,let us now consider how1 and on what account the soul is said to be of a middle nature1 and to be the receptacle of all middle energies1 both vital and gnostic4 ,since1 then1 there is a long gradation of beings1 proceeding from the first being1 even to formless matter1 which is nothing more than the dark shadow of essence1 it is requisite to enquire what the properties are of the first and last beings1 and what the condition is of the middle orders2 for thus we shall know where the essence of the soul ought to be placed4 ,the properties of intelligible natures1 therefore1 are as follows3 true being1 eternal1 indivisible1 immoveable1 total1 perfect1 full of essence1 replete with life1 free1 moving all things1 similitude1 presiding over all things1 and at the same time separated from all2 for each of these properties appears in intelligibles1 according to the processions of being4 ,but the properties of sensible natures1 different from these by the greatest interval1 are such as1 not-true-being1 temporal according to essence1 partible1 moveable1 particular1 indigent of another1 always replete with subsistence1 living by participation1 moved by another1 dissimilitude1 and occupying place by parts4 ,but the middle properties of these are1 not-true-being1 an essence better than non-being1 and inferior to true-being1 according to essence eternal1 but according to its energies extended with time1 indivisible according to its divine part1 but divisible according to the various processions of reasons1 self-motive1 governing things moved by another1 but subordinate to such as are immoveable1 bearing before itself a particular nature1 together with its totality2 7for1 because it contains in itself all reasons1 it is after a manner a whole1 but because it is diminished and fallen1 ends in parts1 and suffers a transition of its energy1 it must be esteemed a particular nature73 and again1 perfecting itself1 yet1 nevertheless1 perfected by natures prior to its own2 filling itself with power and strength1 and at the same time filled by others3 living from itself1 and receiving life from others1 being more divine1 indeed1 than things which live only by participation1 but inferior to things primarily vital2 moving other things1 and itself moved by others2 at the same time similar and dissimilar2 and separated1 at the same time1 from last natures1 and co-ordinated with them4 ,such then1 being the properties of the first1 middle1 and last orders1 let us consider where we ought to place the soul1 whether in the first order1 or in those which retain the last place3 but if we establish it among the first1 it must be true being1 every way eternal and immoveable1 and it must consequently possess every thing which we have attributed to the first beings2 add too1 that on this hypothesis we can no longer attribute to the soul a power of self-motion1 nor the discursive processions of reasoning1 nor a variety of other particulars1 which manifestly belong to the soul4 ,but can we place it with propriety among the last of beings8 ,the least of all3 for on this hypothesis we shall make it alone moved by others1 divisible1 composite1 and alone possessing perfection from others1 the opposite of which is evident in all our souls2 since they move and perfect themselves1 and are led wherever they please4 ,since then it is not possible to place the soul either in the first1 or last order of beings1 it is requisite to assign it a middle place1 in imitation of its divine cause ,rhea 7according to the theologists71 who is the conciliating band of the two parents ,saturn and ,jupiter1 and is reported1 from her prolific bosom to produce the life of the soul4 ,but though the soul is thus the extremity of intelligible1 and the principle of sensible natures1 we must not conceive it to be such a principle or extremity as a point in a line1 for it is not in both the natures it terminates1 like a point in both the sections of a line2 but it is to be called the extremity of intelligibles1 because it appears after an intelligible essence1 and the principle of sensibles1 as being abstracted from them1 and the source of their motion4 ,and thus it will preserve to us a certain proportion1 that as the natures which are moved by others1 are to those which are moved by themselves1 so are these last to immoveable natures2 and hence it will obtain the condition of a bond1 on account of its peculiar mediocrity1 unfolding1 indeed1 united causes1 but reducing the dissipated powers of sensibles into one1 and being contained by an immoveable and perpetually abiding cause2 but containing itself the generation1 which1 moved by another1 is subject to continual mutation4 ,it is likewise intelligible1 if we regard generated natures2 but generated1 if we compare it with intelligibles2 and thus it exhibits in its middle nature both extremes1 imitating also1 by this means1 7according to the ,greek theologists7 its divine cause1 for it is said to be on both sides refulgent1 1 and to be endued with two faces 1 and to receive in its bosom the processions of intelligible natures4 ,it is likewise said to be replenished with intellectual life1 and to be the fountain of the ever-running streams of corporeal life1 and to contain in itself the centre of the processions of all beings4 ,on this account it is1 with great propriety1 affirmed to be generated1 and at the same time without generation4 ,for true being1 according to the ,platonists1 is without generation1 because it has an infinite power of being totally present at the same time3 and body is said to be generated1 because it always possesses in itself an infinite flowing power1 which it cannot at once totally receive4 ,the soul1 therefore1 because it is incorporeal1 abiding in itself1 has an infinite power of being1 and this total with respect to its essence1 and immortal without generation2 but according to parts it may be considered in infinite production4 ,for it has not the same total infinity ever present1 or there would be the same infinity of the whole and part1 of the perfect and imperfect1 of the contained and containing1 which is impossible4 ,but neither is it possible that the whole of its essence should be in the act of perpetual production1 any more than that a part of it should be eternal being1 lest the part should be more worthy and better than the whole4 ,hence the 1 or subsistence of the soul1 is at the same time of infinite power1 and is generated in infinitum2 for by this means it participates of being1 and obtains the first place among generated natures2 while body alone1 both with respect to its whole and parts1 is obnoxious to a perpetual generation4 #c4 ,but let us now enquire from what genera ,plato composes this nature of the soul1 which contains in itself the bond of all beings2 previous to which it will be requisite to explain what these genera are1 and from whence they originate4 ,of the species1 then1 existing in the intelligible world1 or the divine intellect1 which contains in itself the causes of all posterior natures1 some are most general1 extending themselves to the universality of things2 but others are more particular1 like the most special species1 and others subsisting between these1 expand themselves1 indeed1 to a multitude of things1 but not to all1 according to the division1 of the ,elean guest in the ,sophista4 ,for man is produced from the ideal man1 and horse from the ideal horse1 in the intelligible world2 but the similitude which is found in man and horse1 and other animals1 is produced from likeness itself1 or the ideal similitude1 as dissimilitude from unlikeness itself2 but the sameness and difference which are found in all beings1 proceed from the sameness and difference which subsist in ever-vital energy and perfection1 in the supreme intellect1 or the ideal world4 ,now1 as among the sciences some are especially universal1 so in intelligible causes some are perfectly particular1 presiding alone over the proper and peculiar number of one species2 but others extend themselves to a multitude1 such as equality1 likeness1 totality1 7for the whole considered as a whole is not common to all things1 since the part is not a whole72 but others1 again1 expand themselves to all things1 as all beings participate of these1 considered as beings1 and not considered as vital or animated1 or possessing any other property exclusive of the denomination of being4 ,because1 therefore1 being is the first1 the causes of being obtain the most universal order among genera2 and these are five in number1 as follows1 essence1 sameness1 difference1 motion1 and station4 ,for every being is endued with essence2 is united to itself2 is by itself1 or its own sameness1 separated from others2 proceeds from itself1 and its own state and principle1 and no less appears to participate of a certain abiding1 in preserving its own proper species4 ,all things1 therefore1 whether intelligibles or sensibles1 or subsisting between both1 depend on these genera for their existence4 ,for without the being of essence1 nothing could subsist2 in like manner1 without sameness every whole would be dissipated1 and divided from itself2 and difference being taken away1 all things would be one alone1 and multitude be destroyed4 ,but without motion and station1 all things would either be inefficacious and dead1 or1 losing their proper state and stability1 would end in non-entity4 #d4 ,such then being the middle nature of the soul1 ,plato1 with great propriety1 in the ,phdrus1 and in his tenth book of laws1 defines it to be number moving itself2 which definition he received from ,philolaus1 and ,philolaus from ,pythagoras4 ,for since mathematical species have a middle subsistence1 as is proved in the following ,commentaries1 they are of all things most accommodated to the nature of the soul4 ,hence ,plato1 in imitation of ,pythagoras1 sometimes explains the soul by number1 as in the present instance2 and sometimes by figure and magnitude1 as in the ,timus2 while he considers in the soul the intersection of lines1 and a twofold circle4 ,for since mathematical forms are separated from the flux and inconstancy of matter1 they participate of a certain1 exact1 sure1 and exquisite condition1 by means of which they eminently confer to the elevation of our ingenuity1 and the explication of latent concerns2 and1 on this account1 as they pertain to numbers1 we may say1 preserving the analogy1 that there are five orders of numbers1 the divine1 the essential1 the animative1 the natural1 and the mathematic4 ,the first of these is uniform1 the second immoveable1 the third self-motive1 the fourth moved by another1 and the last the image of the others1 and their external measure4 ,the divine number is considered eminently in the deity1 as in the principle of all things2 the essential belongs to intellect through ideas1 and is called essence1 unity1 and the first being2 the animative number belongs to the soul1 through the medium of her inherent reasons2 the natural to physical concerns1 through the seeds of nature2 and lastly1 the mathematical belongs to opinion1 as it is nothing more than the image of essential number1 formed by the energies of the rational soul4 ,the soul1 therefore1 is number1 not limited by quantity1 and mathematical1 but animative2 it is number1 not indeed numbering1 but numbered1 generating and converted into itself4 ,hence too1 because harmony arises from number1 the soul is called harmony2 not1 indeed1 a harmony of the parts of the body1 nor the harmonic quantity which subsists in sound or in voice2 but a harmony arising from its essential numbers1 placed in its inherent reasons1 and in the genera which constitute its nature4 ,it is this harmony which produces1 as from its proper cause1 the harmony of the corporeal parts1 the rhythm of motions1 and the melody of voices and sounds4 ,it is this which produces that delight in the soul from sensible harmony1 which sufficiently indicates it to be something familiar and domestic to her nature4 ,from hence it may be inferred1 that ,plato is not inconsistent with himself when1 in the ,phdo1 he denies1 and in the ,timus affirms1 the soul to be harmony2 for he denies that it is a harmony of a definite quantity1 or such as arises from the parts of the body2 but he asserts it to be a harmony in the manner already explained4 ,it may likewise be inferred1 that ,plato is ignorantly accused by a many1 for affirming that the soul is harmony1 or number2 for they only regard vulgar mathematical number1 and sensible harmony2 while ,plato1 far more elevated1 discourses of intelligible numbers1 and ideal harmony1 subsisting in immaterial energy and perfection4 #e4 ,and here it is necessary to consider what number1 in a particular manner belongs to the soul2 for various numbers1 differently considered1 accord with her self-motive nature4 ,in the first place1 union and unity may be considered in the soul1 as in her proper degree she participates of divine unity2 and likewise with relation to her totality1 for she is one certain whole4 ,and because a whole may be considered in a triple respect1 one before the parts1 another rising from the aggregate of parts1 and a third subsisting in the single parts2 the soul is a whole in each of these respects4 ,thus she is a whole prior to the parts1 while she is considered as divisible into them1 in an incorporeal manner2 she is a whole rising from parts1 while assuming the parts in the first place1 we consider how her nature is fabricated from their conjunction2 and she is a whole in the single parts1 since she is total in the whole1 and in every part4 ,besides1 the duad belongs to the soul1 because she contains in her nature bound and infinite1 sameness and difference2 and lastly1 a conversion to intelligibles and sensibles4 ,and1 indeed1 the duad conjoined with unity1 very properly accords with the soul2 for to intellect above soul1 unity particularly belongs2 to body beneath soul1 the infinite alone2 and to soul situated in the middle1 duality properly agrees1 being1 as it were1 infinite1 connected with unity4 ,again1 the ternary number is attributed to the soul1 as well on account of her beginning1 middle1 and end1 as because she abides in herself1 proceeds to inferiors1 and returns to supernal natures4 ,besides1 she flows from the one1 recedes from him1 and is reflected into the one when she acquires her proper perfection4 ,lastly1 as ,proclus observes1 the nature of the soul is divided into essence1 power1 and energy2 so that she may be said to rejoice in the ternary number1 and to be replete with its perfection4 ,but the quaternary number belongs to the soul so far as she is connected with matter1 which is tempered with four qualities1 and four elements2 and she is endued with four principal faculties1 nutrition1 sensation1 local motion1 and intellection4 ,but to omit other numbers1 and their conformity with the soul1 the quinary1 and septenary numbers are especially attributed to the soul4 ,the quinary1 because the soul is composed from the five genera of things1 we have previously explained2 and because five particulars merit a principal consideration in the soul2 first1 her essence2 secondly1 the harmony of her reasons2 thirdly1 the species arising from the concord of her parts or reasons2 fourthly1 her virtue2 and1 lastly1 her energies3 and on this account1 ,proclus observes1 the consideration of the soul ought to receive a quintuple distribution4 ,besides1 as the soul consists from a divisible and indivisible nature2 so the quinary is composed from the first even and the first odd number4 ,lastly1 as the soul is the connecting medium of the universe2 so the quinary obtains the middle situation in universal number1 that is in the decad4 ,but the septenary number belongs to the soul1 because1 as ,plato shews1 in his ,timus1 all harmonical reasons are contained in the seven numbers1 #a1 #b1 #c1 #d1 #i1 #h1 #bg2 and from these the soul is composed4 ,hence ,proclus elegantly observes1 that the septenary number is dedicated to ,apollo1 the parent of all harmony2 because in one1 two1 and four1 from which the septenary results1 the first bisdiapason is found4 ,besides1 the writers on harmony affirm that all the difference of voices proceeds as far as to the seventh degree4 ,so that ,plato uses1 with great propriety1 the septenary number for the composition of the soul4 ,again1 in these numbers of the soul1 every medium is found in a convenient proportion2 in these the geometric medium is contained1 corresponding to the right institution of laws2 of which ,plato1 in his ,republic1 says1 that by this cities are properly governed3 there is found too1 an harmonical medium1 which is the similitude of justice3 and lastly1 we may discover an arithmetical medium1 which is called the illustrious symbol of peace4 ,after this manner1 therefore1 ,plato1 with a wonderful fecundity of significations1 affirms that the soul is composed from numbers4 ,he likewise considers the figures she contains1 ,i mean the circle and triangle2 because as the soul is the first nature which verges to body1 so these are the first of all figures1 as well rectilinear as curvilinear4 ,besides1 an orbiculation agrees to the soul1 through intellect2 but progression and rectitude according to her own proper nature2 and on this account she contains both a straight and circular figure4 ,i omit other correspondencies of a circle and triangle with the soul1 as they are exhibited in the following ,commentaries2 and particularly by ,proclus1 in the third book of his ,commentaries on the ,timus3 it is sufficient to the design of this ,dissertation1 just to have mentioned this analogy1 that the nature of the numbers and figures may appear1 which are considered by ,plato in the composition of the soul4 #f4 ,let us now pass from contemplating the nature of the soul1 to a survey of its various gradations of knowledge1 and the means by which it acquires the illuminations of science2 as this is a speculation perfectly essential to a full comprehension of the ensuing ,commentaries4 ,according to ,plato1 then1 in the sixth and seventh books of his ,republic#bh1 there are four degrees of the internal cognitions of our soul2 imagination1 or assimilation2 faith1 cogitation 772 and lastly1 science or wisdom4 ,the two first degrees conjoined constitute opinion2 but the two last equally joined produce intelligence in its large acceptation4 ,i say in its large acceptation1 because the word intelligence is considered by the ,platonists in a triple respect4 ,first1 as it rises from opinion and science2 as ,plato asserts1 in the seventh book of his ,republic4 ,secondly1 as it passes into the same with science2 for thus1 in the end of the sixth book1 he considers intelligence and science as the same4 ,lastly1 as it is distinguished from science1 and intelligible from that which is the object of cognition3 thus1 science regards the essential reasons of the soul2 but intelligence elevates us to ideas1 and this is intelligence in its proper acceptation4 ,the distinction of these four degrees1 especially depends on the distinction of things with which the soul is conversant4 ,for these four degrees of things are1 the image of a sensible object1 the sensible object itself1 the image of an intelligible object1 and the intelligible itself4 ,imagination or assimilation regards the image of the sensible object2 which image is nothing more than the shadow or resemblance appearing in water1 or other lucid and polished bodies4 ,faith is conversant with sensible objects2 and these are animals1 plants1 and every thing subject to the energies of sense4 ,from the junction of imagination and faith1 opinion is produced4 ,cogitation is conversant with the image of an intelligible object2 which is nothing more than a certain universal1 collected from sensibles1 related to the reasons of things existing in the soul1 and constituted by their assistance2 but not elevated to ideas1 and resolved into their lucid nature4 ,those who are conversant with this image of an intelligible object1 use hypotheses2 which do not elevate us to principles1 which are reasons and ideas1 but bring us down to subordinate objects4 ,lastly1 science1 considered as the same with intelligence1 is conversant with that which is intelligible1 or the essence of things2 and of this kind are reasons pertaining to the soul1 and ideas to intellect4 ,hence1 as intelligible is to sensible in splendor and truth1 and form to its image2 such is the relation of the superior to the inferior degrees of cognition4 ,and as these four degrees may be resolved into five1 by separating intelligence from science2 so they are reduced by ,plato into two principal degrees2 so far as the two first are conversant about generation2 but the two last about essence4 ,and thus much for the first particular proposed4 ,it now remains that we investigate the mode in which science is produced in the soul1 according to the doctrine of ,plato4 ,for this purpose1 ,i think it will be necessary to consider the rational soul1 in whose nature all reasons corresponding to ideas1 have been inserted from eternity1 received into the human body1 as into the plain of oblivion2 bordering on the river of negligence1 that is placed near to the flux of humours2 which producing in its nature various kinds of perturbations1 are the causes of its self-oblivion and neglect4 ,hence the soul1 thus constituted1 as it were1 sleeping and intoxicated2 sleeping before it is roused2 intoxicated before it is purified2 begins from external sensible objects to be moved and excited1 and to seek with avidity1 the knowledge congenial to its nature4 ,the soul now1 enquiring after truth and the sciences1 is first conversant with the external images of things1 in which the glimmering light of similitude to truth1 presents itself to the view2 then it distinguishes these images among themselves by its reasoning power2 and if they agree in any particular1 collects them into one4 ,thus1 being employed in separating into many1 things united1 and reducing many into one1 it advances from shade to substance1 and is elevated from similitude to truth itself2 and thus apprehends the essence of a thing free from every foreign or contrary quality1 shining in reason and idea4 ,afterwards1 from this contact1 especially salutary to itself1 it experiences an ineffable joy1 as from a return into its proper nature1 and best disposition2 and so great is its exultation1 that it neglects and despises the shadows which it formerly pursued4 ,then the soul truly knows1 that while a man regards corporeal natures1 he is employed in resemblances2 and that though he may esteem himself knowing in many things1 he knows nothing in reality2 but is then alone elevated to the sublime degree of science1 when he arrives at ideas4 ,hence it appears1 that there are four subordinate dispositions of mankind in order to science4 ,for1 in the first place1 children1 as new guests of this world1 are ignorant of every thing1 without being conscious of their ignorance4 ,but as they advance in years1 they are employed in the shadows and images of natural concerns2 and not being yet converted to the essences of things1 though they possess no real knowledge1 yet they conceive themselves to abound in knowledge4 ,but in the third place1 being elevated to supernal natures1 and judging these to be alone true1 they affirm themselves to be ignorant of all they formerly imagined themselves to know2 in which degree ,socrates professed to find himself1 when he said1 this one thing ,i know1 that ,i know nothing2 an ignorance preferable to all the knowledge gained by the most unwearied experimental enquiries4 ,indeed1 this order is indicated by ,socrates himself1 in the ,phdo1 when he says1 that on his first acquaintance with natural concerns1 he thought he abounded in knowledge1 but as he advanced in these pursuits1 he perceived that he knew nothing4 ,lastly1 when1 through the piercing sight of the soul1 men are elevated to ideas1 they become illustrated with the splendors of true science1 and pass into the regions of perfect reality4 ,and hence it appears how true science1 which is the same with intelligence and wisdom1 produces piety and religion3 for science elevates us to intellect and divinity2 and copulates the soul with natures of the greatest purity and perfection2 so that an union of this kind cannot take place without piety1 sanctity1 and religion2 as dissimilars can never be blended in amicable conjunction4 ,,section ,,iv4 ,we are informed by ,proclus1 in the ensuing ,commentaries1 that the end of geometry1 and1 indeed1 of mathematics in general1 is to be referred to the energies of intellect2 and that it is degraded when made subservient to the common utilities of a mere animal life4 ,but as the very opposite to this is the prevailing opinion of the present age1 let us examine the truth of this doctrine1 and attend to the arguments which the ,platonic philosophy affords in its defence4 ,for if we can prove that this assertion of ,proclus is supported by the strongest evidence1 we shall vindicate the dignity of true geometry1 restore it to its ancient esteem in the minds of the liberal1 and shew how much it is perverted by applying it to contrary purposes4 ,in order to this1 ,i shall endeavour to prove the following position1 that things valuable for their own sakes1 are preferable to such as refer to something else4 ,now1 this may be demonstrated1 by considering that every natural production was made with reference to some end1 as is evident from an induction of particulars2 and if this be the case1 it may be safely inferred1 that every thing exists for the sake of the end4 ,but that for the sake of which any being subsists is the best of all2 and the end1 according to nature1 is that which is perfected the last of all1 from the birth of any being4 ,hence the human body receives its end or perfection first1 but the soul last4 ,and hence the soul is posterior to the body1 in the accomplishment of its nature2 and its ultimate perfection is wisdom4 ,it is on this account that old age alone pursues and desires the goods of prudence and wisdom4 ,hence1 wisdom is a certain end to us according to nature2 and to be wise1 is the extreme or final cause for which we were produced4 ,it was1 therefore1 beautifully said by ,pythagoras1 that man was constituted by divinity1 that he might know1 and contemplate4 ,if then wisdom be the end of our nature1 to be endued with wisdom must be the best of all4 ,so that other things are to be performed for the sake of the good which this contains4 ,but to enquire in every science something besides this1 and to require that it should be useful1 is alone the employment of one ignorant of the great difference between the most illustrious goods1 and things necessary4 ,for they differ1 indeed1 widely2 since things are to be called necessaries1 which are the objects of desire for the sake of others1 and without which it is impossible to live4 ,but those concerns alone are properly good1 which are loved by themselves1 though nothing else should fall to the lot of their possessor2 for one thing is not to be desired for the sake of another infinitely1 but it is requisite to stop at some limited object of desire1 of which it would be ridiculous to require any utility abstracted from itself4 ,but you will ask1 ,what is the emolument of contemplative wisdom1 what the good it confers on its possessor8 ,what if we should say 7for such is the truth of the case7 that it transports us by intellect and cogitation1 to regions similar to the fortunate islands2 for utility and necessity are strangers to those happy and liberal realms4 ,and if this be admitted1 ought we not to blush1 that having it in our power to become inhabitants of the fortunate islands1 we neglect the pursuit1 through a sordid enquiry after what is useful and profitable according to vulgar estimation8 ,the rewards of science1 therefore1 are not to be reprehended1 nor is it a trifling good which results from its acquisition4 ,besides1 as men travel to the mountain ,olympus for its spectacle alone1 preferring a view of its lofty summit to much wealth2 and as many other spectacles are desired for their own sakes1 and valued beyond gold1 in like manner the speculation of the universe is to be prized above every thing which appears useful to the purposes of life3 for it is surely shameful that we should eagerly frequent the theatre1 and the race1 for the sake of the delight afforded to our corporeal sight1 and should look for no farther utility in these than the pleasure they produce2 and yet should be so sordidly stupid as to think that the nature of things1 and truth itself are not to be speculated without some farther reward than the sincere delight their contemplation affords4 ,it is on this account that the apprehension of truth is compared to corporeal vision2 for the sight is the most liberal of all the senses1 as is confirmed by the general testimony of mankind4 ,hence1 the sight of the sun and moon1 and the glorious spectacle of the stars is desired by the most illiterate as well as the most knowing1 for the delight such visions afford2 while1 on the contrary1 the desires of the other senses are for the most part directed to something farther than the mere objects of their energy4 ,thus1 even the sense of hearing1 which is the next in dignity to the sight1 is not always desirable for its own sake2 for light is the general object of sight1 and sound that of hearing2 but it is evident that light is more universally desired than sound1 since all light1 when not excessive1 is always pleasing1 but this is by no means the case with every kind of sound4 ,hence it is1 that all contemplation is so delightful1 and this in proportion as it becomes abstracted from sensible objects2 for the most beautiful forms do not produce genuine delight1 until they are strongly represented in the phantasy1 as is evident in the passion of love2 since the fairest face then alone causes love when it presents itself clearly to the inward eye of thought1 in the mirror of imagination1 accompanied with living elegance1 and a resistless energy of form4 ,indeed1 so liberal and so exalted an employment is contemplation1 that ,plotinus1 with his usual profundity1 proves that the universe subsists for its sake2 that all the productions of nature originate from this2 and that even actions themselves are undertaken with a view to the enjoyment of after-speculation4 ,may we not1 therefore1 say that the sportsman follows the chace for the sake of a subsequent review of his favourite pursuit8 ,that the glutton for this rejoices in the meal2 and even the miser in his wealth8 ,and that conversation is alone sollicited1 that it may recal past images to the soul8 ,in short1 contemplation is the first spring of action1 and its only end2 since we are first incited to any external object by speculating its image in the phantasy3 and our subsequent conduct tends1 without ceasing1 to the energy of reflection2 for destroy prior and posterior contemplation1 and action is no more4 ,now if this be the case1 and if geometry is a speculative science 7,i mean the geometry of the ancients71 it is both desirable for its own sake1 and for still higher contemplations1 the visions of intellect1 to which it is ultimately subservient4 ,for1 when studied with this view1 it opens the eye of the soul to spectacles of perfect reality1 and purifies it from the darkness of material oblivion4 ,away then1 ye sordid vulgar1 who are perpetually demanding the utility of abstract speculations1 and who are impatient to bring down and debase the noblest energies1 to the most groveling purposes2 ignorant of that mighty principle of action1 which influences every part of the universe1 and through which even division and discord tend as much as possible to union and consent2 ignorant that from the depravity of your nature1 and the blindness of your inward eye1 you are incapable of speculating the substance of reality1 and are therefore eagerly gazing on its shadow3 and lastly1 unconscious that this is the point about which you are continually making excentric revolutions1 mistaking the circumference for the centre1 motion for rest1 and a departure from good for a tendency to felicity4 ,it was for the sake of this most exalted and liberal contemplation that ,heraclitus yielded his right of succession to a throne1 to his brother2 and that ,anaxagoras neglected his patrimony1 esteeming one drop of genuine wisdom preferable to whole tuns of riches4 ,led by a desire of this1 as by some guiding star1 ,pythagoras travelled into ,egypt1 and cheerfully encountered the greatest difficulties1 and maintained the most obstinate perseverance1 until at length he happily penetrated the depths of ,egyptian wisdom1 and brought into ,greece a treasury of truth for future speculation4 ,but these were happy days2 this was the period destined to the reign of true philosophy1 and to the advancement of the human soul to the greatest perfection its union with this terrene body can admit4 ,for in our times1 the voice of wisdom is no longer heard in the silence of sacred solitude2 but folly usurping her place1 has filled every quarter with the barbarous and deafning clamours of despicable sectaries2 while the brutal hand of commerce has blinded the liberal eye of divine contemplation4 ,for unfortunately1 the circle of time1 as it produces continual variations1 at length reverses the objects of pursuit2 and hence1 that which was once deservedly first1 becomes at length1 by a degraded revolution1 the last in the general esteem4 #b4 ,if geometry1 therefore1 be both valuable for its own sake1 and for its subserviency to the most exalted contemplations1 there can be no doubt but that the great perfection to which this science was brought by the ,greeks1 was entirely owing to their deep conviction of this important truth4 ,euclid1 we are informed by ,proclus1 in this work1 was of the ,platonic sect2 and ,archimedes is reported1 by ,plutarch1 in his ,life of ,marcellus1 to have possessed such elevated sentiments of the intrinsic dignity of geometry1 that he considered it perverted and degraded1 when subservient to mechanical operations2 though1 at the request of king ,hiero1 he fabricated such admirable engines for the defence of ,syracuse4 ,from this source alone1 the great accuracy and elegance of their demonstrations was derived1 which have been so deservedly applauded by the greatest modern mathematicians1 and the warmest advocates for the farrago of algebraic calculation4 ,algebra1 indeed1 or as it is called1 specious analysis1 is the modern substitute for the perfect method adopted by the ancients in geometrical demonstrations2 and this solely1 because it is capable of being applied with greater facility to the common purposes of life4 ,hence1 hypotheses have been eagerly admitted in geometry1 which the ancients would have blushed to own3 ,i mean the multiplications and divisions of lines and spaces as if they were numbers1 and considering geometry and arithmetic as sciences perfectly the same4 ,but we have fortunately the testimony of the first mathematicians among the moderns against the unlawfulness of this ungeometrical invasion4 ,and to begin with the great sir ,isaac ,newton1 in his ,universal ,arithmetic#bi3 8,equations 7says he7 are expressions of arithmetical computation1 and properly have no place in geometry1 except so far as quantities truly geometrical 7that is1 lines1 surfaces1 solids1 and proportions71 may be said to be some equal to others4 ,multiplications1 divisions1 and such sort of computations1 are newly received into geometry1 and that unwarily1 and contrary to the first design of this science4 ,for whoever considers the construction of problems by a right-line and a circle1 found out by the first geometricians1 will easily perceive that geometry was invented that we might expeditiously avoid1 by drawing lines1 the tediousness of computation4 ,therefore1 these two sciences ought not to be confounded4 ,the ancients so industriously distinguished them from one another1 that they never introduced arithmetical terms into geometry4 ,and the moderns1 by confounding both1 have lost the simplicity in which all the elegancy of geometry consists40 ,and in another part#cj of the same work he observes1 that 8the modern geometers are too fond of the speculation of equations40 ,to this very high authority we may add that of ,dr4 ,halley1 in the preface to his translation of ,apollonius de ,sectione ,rationis2 for which work he conceived so great an esteem1 that he was at the pains to learn ,arabic in order to accomplish its translation into ,latin#ca3 8,this method1 says he1 7of ,apollonius7 contends with specious algebra in facility1 but far excels it in evidence and elegance of demonstrations2 as will be abundantly manifest if any one compares this doctrine of ,apollonius de ,sectione ,rationis1 with the algebraic analysis of the same problem1 which the most illustrious ,wallis exhibits in the second volume of his mathematical works1 cap4 liv4 p4 #bbj40 ,and in the conclusion of his preface1 he observes#cb1 8that it is one thing to give the resolution of a problem some how or other1 which may be accomplished by various ways1 but another to effect this by the most elegant method2 by an analysis the shortest1 and at the same time perspicuous2 by a synthesis elegant1 and by no means operose40 ,and ,dr4 ,barrow1 notwithstanding he was so great an advocate for the identity of arithmetic and geometry1 expressly asserts#cc1 that algebra is no science4 ,to these authorities we may add ,simson and ,lawson1 who1 sensible of the superior skill of the ancients1 both in analysis and synthesis1 have made laudable attempts to restore the ,greek geometry to its pristine purity and perfection4 ,again1 the greatest men of the present times have been of opinion1 that algebra was not unknown to the ancients2 and if this be true1 their silence respecting it is a sufficient proof of their disapprobation4 ,indeed1 if we consider it when applied to geometry1 as an art alone subservient to the facility of practice1 as conveying no evidence1 and possessing no elegance of demonstration1 we shall not wonder at its being unnoticed by the ancients1 with whom practice was ever considered as subservient to speculation2 and in whose writings elegance of theory and accuracy of reasoning are found perpetually united4 #c4 ,but the lives of the first cultivators of this science 7,i mean the ,egyptian priests7 as well as of the ,pythagoreans and ,platonists1 by whom it afterwards received such improvements1 sufficiently evince that this science advanced to perfection from an intellectual theory as its source1 and from being referred to contemplation as its end2 and this will be evident1 by attending to the following history of the ,egyptian priests1 as preserved to us by ,porphyry1 in his excellent work on abstinence#cd2 a translation of which will not1 ,i presume1 be unacceptable to the philosophical reader1 8,chremon1 the ,stoic 7says he7 explaining the rites of the ,egyptian priests1 who1 he says1 are accounted philosophers by the ,egyptians1 relates1 that they choose a place best adapted to the study and performance of sacred rites2 so that a desire of contemplation is excited by only frequenting those recesses which are dedicated to their use1 and which procure safety to the priests1 on account of that reverence of the divinity1 whose sacred mysteries they perform2 so that all possible honour is paid to these philosophers1 in the same manner as to some sacred animals4 ,but he says they live entirely solitary1 except at particular times1 when they mix with others in such assemblies as are usually held1 and in public feasts2 and that on all other occasions they are scarcely to be approached4 ,for he who desires to converse with them must first purify himself1 and abstain from a multitude of things after the manner of these ,egyptian priests4 ,he adds1 that these men1 renouncing every other occupation1 and all human affairs1 give themselves entirety1 through the whole of life1 to the contemplation of divine concerns1 and to enquiring into the divine will3 by the latter of these employments procuring to themselves honour1 security1 and the estimation of piety2 by contemplation1 tracing out the latent paths of science2 and by both these occupations united1 accustoming themselves to manners truly occult1 and worthy of antiquity4 ,for to dwell always on divine knowledge1 and be disposed for divine inspiration1 removes a man beyond all immoderate desires1 calms the passions of the soul1 and raises her intellectual eye to the perception of that which is real and true4 ,but they studied tenuity of aliment1 and frugality in their apparel1 and cultivated temperance and patience1 together with justice and equity1 in all their concerns4 ,indeed1 a solitary life rendered them perfectly venerable2 for during that period which they call the time of purification1 they scarcely mixed with the associates of their own order1 or saw any one of them1 except him who was conversant with them in that exercise of purity1 on account of necessary uses4 ,but they by no means concerned themselves with those who were unemployed in the business of purification4 ,the remaining part of their time they conversed familiarly with those similar to themselves2 but they lived separate and apart from those who were estranged from their ceremonies and manner of living4 ,he adds1 they are always seen employed among the resemblances of the gods1 either carrying their images1 or preceding them in their accustomed processions1 and disposing them with gravity of deportment1 and in a graceful order4 ,in all which operations they did not indicate any pride of disposition2 but exhibited some particular natural reason4 ,but their gravity was conspicuous from their habit2 for when they walked1 their pace was equable1 and their aspect so perfectly steady1 that they refrained from winking whenever they pleased4 ,their risibility too1 extended no farther than to a smile4 ,but their hands were always contained within their garments2 and as there were many orders of priests1 every one carried about him some remarkable symbol of the order he was allotted in sacred concerns4 ,their sustenance too was slender and simple2 and with respect to wine1 some of them entirely refrained from it2 and others drank it very sparingly1 affirming that it hurt the nerves1 was an impediment to the invention of things1 and an incentive to venereal desires4 ,they also abstained from many other things1 never using bread in exercises of purity2 and if they ate it at other times1 it was first cut in pieces1 and mingled with hyssop4 ,but they abstained1 for the most part1 from oil1 and when they used it mixt with olives1 it was only in small quantities1 and as much as was sufficient to mitigate the taste of the herbs4 ,in the mean time1 it was not lawful for any one to taste of the aliment1 whether solid or fluid1 which was brought into ,egypt from foreign parts4 ,they likewise abstained from the fish which ,egypt produced2 and from all quadrupeds having solid or many fissured hoofs2 from such as were without horns2 and from all carniverous birds3 but many of them abstained entirely from animal food4 ,and at those times when they all rendered themselves pure1 they did not even eat an egg4 ,but when the time drew near in which they were to celebrate some sacred rites1 or festival1 they employed many days in previous preparation1 some of them setting apart forty-two days1 others a greater length of time than this2 and others again a shorter2 but never less than seven days2 abstaining1 during this period1 from all animals1 and from all leguminous and oily nutriment1 but especially from venereal congress4 ,every day1 they washed themselves three times in cold water2 after rising from bed1 before dinner1 and when they betook themselves to rest4 ,and if they happened to be polluted in their sleep1 they immediately purified their bodies in a bath4 ,they made cold water too subservient to the purposes of purification at other times1 but not so often as the bath4 ,their beds were composed from the branches of palm1 which they called 1 bais4 ,a piece of wood1 of a semi-cylindrical form1 and well planed1 served them for a pillow4 ,but through the whole of life1 they were exercised in the endurance of hunger and thirst1 and accustomed to a paucity and simplicity of nutriment4 ,but as a testimony of their temperance1 though they neither used the exercise of walking nor riding1 yet they lived free from disease1 and were moderately strong4 ,for1 indeed1 they endured great labour in their sacred ceremonies1 and performed many services exceeding the common strength of men4 ,they divided the night between observations of the celestial bodies1 and offices of purity2 but the day was destined by them to the cultivation of the divinities1 whom they worshiped with hymns each day three or four times2 in the morning and evening1 when the sun is at his meridian1 and when he is setting4 ,but the rest of their time they were occupied in arithmetical and geometrical speculations1 always laborious and inventing1 and continually employed in the investigation of things4 ,in winter nights also1 they were diligent in the same employments1 and were ever vigilant to literary studies2 since they were not solicitous about external concerns1 and were freed from the base dominion of intemperate desires4 ,their unwearied and assiduous labour1 therefore1 is an argument of their great patience2 and their continence is sufficiently indicated by their privation of desire4 ,besides this1 it was esteemed very impious to sail from ,egypt1 as they were particularly careful in abstaining from the manners and luxuries of foreign nations2 so that to leave ,egypt was alone lawful to those who were compelled to it by state necessities4 ,but they discoursed much concerning a retention of their native manners2 and if any priest was judged to have transgressed the laws in the least particular1 he was expelled from the college4 ,besides1 the true method of philosophizing was preserved in ,commentaries and ,diaries1 by the prophets and ministers of sacred concerns3 but the remaining multitude of priests1 ,pastophori1 or priests of ,isis and ,osiris1 governors of temples1 and servants of the gods1 studied purity1 yet not so exactly1 nor with so great continence as those we have mentioned4 ,and thus much is related of the ,egyptians1 by a man who is equally a lover of truth1 and of accurate diligence1 and who is deeply skilled in the ,stoic philosophy40 #d4 ,but the lives of the ,pythagoreans and ,platonists1 who carried this divine science to its ultimate perfection1 no less eminently evince the truth of our position4 ,for1 as ,porphyry informs us1 in the same invaluable treatise#ce1 8some of the ancient ,pythagoreans1 and wise men1 inhabited the most desert places2 and others retired into temples1 from which the multitude and every tumult were expelled4 ,but ,plato was willing to fix his academy in a place not only solitary1 and remote from the city1 but1 as they report1 insalubrious4 ,others1 again1 have not spared their eyes1 through a desire of more perfectly enjoying that blissful contemplation1 from which they wished never to be separated40 ,after this1 he presents us with a description from ,plato#cf of those intellectual men1 by whom the world has been enlightened with the sublimest wisdom and truth3 8,for it was not falsly1 or in vain 7says he71 that a certain philosopher1 speaking of contemplative men1 affirms1 that such as these are ignorant1 from their early youth1 of the way which leads to the forum1 or in what place the court or senate-house is situated1 or any public council of the state4 ,they neither see nor hear the laws1 whether decreed or promulgated1 or written2 and with respect to the factions and contentions of their companions for magistracy1 for assemblies and splendid entertainments1 luxurious eating and minstrels1 they do not even think of these as in a dream4 ,such an one knows no more of the evil which has happened to some one of his ancestors1 whether male or female1 or any thing belonging to them1 than how many pitchers of water are contained in the sea4 ,nor does he abstain from things of this nature for the sake of acquiring fame2 but in reality1 his body alone abides in the city1 and wanders about from place to place1 but his intellect esteeming all these as of small importance1 or rather as non-entities1 he despises them1 and1 according to ,pindar1 8from these on every side he soars30 by no means applying himself to things which are near him1 and to sensible concerns40 ,if such then were the lives of the men who brought this contemplative science to its present perfection1 and who are to this day our masters in geometry2 if such were the exalted sentiments they entertained of its dignity and worth1 what greater proof can we require of its being valuable for its own sake1 and as subservient to the energies of intellect8 ,we have ample evidence too1 of its being degraded when brought down to the common purposes of life1 in the example of those who1 with this view1 have disguised it with the dark and sordid involutions of algebraic calculation2 for it was solely to facilitate practice1 that this barbarous invasion has been admitted by the moderns4 ,let me then be permitted to persuade the few who study geometry in its ancient purity1 and who consider the ruins of ,grecian literature on this1 as well as on every other science1 the models of perfection1 to enter with avidity on the study of the ensuing ,commentaries1 and endeavour to fathom the depth of our profound and elegant philosopher3 for by this means they may happily obtain the end of all true science1 the purification of the soul2 and be able to draw the light of perfect wisdom1 from the undecaying and inexhaustible fountain of good4 ,but if it should be asked in what these energies of intellect consist1 to which all science ultimately refers8 ,i answer1 in the contemplation of true being1 or those ideal and divine forms1 with which the intelligible world is replete4 ,now this great end is not to be accomplished without previous discipline1 a long exercise of the reasoning power1 and a continued series of philosophic endurance4 ,for this end1 when attained1 is no other than the enjoyment of that felicity congenial to the soul previous to her immersion in body4 ,but1 for the further information of the liberal reader on this important subject1 the following paraphrases from ,porphyry and ,proclus are subjoined2 the former instructing us in the various purifications necessary to this end2 and the latter exhibiting the gradations by which we may rise to the speculation of reality1 and 7leaving all multitude behind7 ascend to the divinely solitary principle of things1 the ineffable ,one4 #e4 8,in the first place1 then 7says ,porphyry#cg7 my reasons are not addressed to those who are occupied in illiberal arts1 nor to those engaged in corporeal exercises1 neither to soldiers nor sailors1 neither to rhetoricians nor to those who have undertaken the duties of an active life4 ,but ,i write to the man continually employed in thinking what he is1 from whence he comes1 and whither he ought to tend3 and who1 with respect to every thing pertaining to food1 and other offices of life1 is entirely changed from those who propose to themselves a different manner of living2 for to a man of this kind alone is my present discourse addressed4 ,indeed1 in this common state of existence1 one and the same mode of persuasion cannot be addressed to the sleeper1 who1 if it was possible1 would conciliate to himself perpetual sleep1 and who1 for this purpose1 seeks on every side for soporiferous incentives1 as to him who studies continually to drive away sleep1 and to dispose every thing about him to vigilance and intellectual activity4 ,but to the former1 it is necessary to advise intoxication1 surfeiting1 and satiety1 and to recommend a dark house2 and1 as the poets say1 a bed luxurious1 broad1 and soft4 ,such a one should chuse whatever tends to produce stupor1 and give birth to indolence and oblivion1 whether consisting of odours1 ointments1 or medicaments which are accustomed to be eat or drank4 ,but it is necessary that the intellectual man should use sober drink1 unmixed with the lethargic fumes of wine2 nutriment slender1 and almost approaching to fasting2 a lucid house1 receiving a subtle air and wind2 that he should be continually agitated with cares and griefs2 and lastly1 that he prepares for himself a small and hard bed1 while thus employed in purifying his soul from the stains contracted by corporeal involution4 ,but whether we are born for this exalted purpose1 ,i mean for vigilant intellectual energies1 allowing as small a part of our life as possible to sleep2 7since we do not exist in a place where souls perpetually vigilant abide71 or whether we are destined to a contrary purpose1 ,i mean1 to sleep and oblivion1 would be foreign from our design to explain2 and would require a longer demonstration than the limits of our work will admit4 ,but whoever once cautiously surmizes the delusions of our life in the present world1 and the inchantments of this material house in which we are employed1 and who perceives himself naturally adapted to vigilant energies2 lastly1 who apprehends the soporiferous nature of the place in which he acts1 to such a one we would prescribe a diet congruous to his suspicion of this fallacious abode1 and to the knowledge he possesses of himself2 in the mean time1 advising him to bid a long farewel to the sleeper1 stretched on his couch1 as on the lap of oblivion4 ,nevertheless1 we should be careful lest1 as those who behold the bleer-eyed1 contract a similar defect1 and as we gape when present with those who are gaping1 so we should be filled with drowsiness and sleep1 when the place in which we reside is cold1 and adapted to fill the eyes with watery humours1 from its abounding with marshes and vapours1 which incline their inhabitants to heaviness and sleep4 ,if then1 legislators had composed the laws with a view to the utility of the state1 and had referred these to a contemplative and intellectual life as their end1 we ought to submit to their institutions1 and acquiesce in the diet they have prescribed for our subsistence4 ,but if they1 only regarding that life which is according to nature1 and is called of the middle kind1 ordain such things as the vulgar admit1 who only estimate good and evil as they respect the body1 why should any one1 adducing these laws1 weary himself in endeavouring to subvert a life which is far more excellent than every law written and composed for the sake of the vulgar1 and which follows a law not written1 but divinely delivered8 ,for such is the truth of the case4 ,that contemplation which procures us felicity1 is not a mass of discourses1 and a multitude of disciplines2 or1 as some may think1 consisting from hence2 nor does it receive any increase from a quantity of words4 ,for if this was the case1 nothing could hinder those from being happy1 who comprehend all disciplines1 and are accurately skilled in a variety of languages4 ,but the whole circle of the sciences cannot by any means accomplish this blissful contemplation1 nor even those disciplines which are conversant with true and substantial being1 unless there is also a conformation of our nature and life to this divine end4 ,for since there are1 as they say1 three ends of living1 if we regard the particular objects to which mankind tend1 the end with us is to follow the contemplation of true being1 promoting1 as much as possible1 by an acquisition of this kind1 an intimate union of the contemplating individual with the object of contemplation4 ,for1 in nothing else besides true being1 is it possible for the soul to return to its pristine felicity2 nor can this be effected by any other conjunction4 ,but intellect is true being itself3 so that the proper end is to live according to intellect4 ,and on this account1 exoteric discourses and disciplines1 retarding the purgation of the soul1 are far from filling up the measure of our felicity4 ,if then1 felicity was defined by the comprehension of words or sciences1 they who do not pay a proper attention to the kind and quantity of their food1 nor to any thing else pertaining to their present existence1 might obtain this end3 but since it is requisite to change our life1 and to be pure both in speech and action1 let us consider what discourses and what works may render us partakers of this most necessary means of acquiring substantial felicity4 ,are1 then1 those things which separate us from sensible objects1 and from the affections which they excite1 and which lead to a life intellectual1 and void of imagination and passion1 are these the means we are in pursuit of8 ,so that every thing contrary is foreign from our purpose1 and worthy to be rejected8 ,and in such proportion as it draws us aside from intellect8 ,indeed1 ,i think it is consonant to truth1 that we should eagerly contend where intellect leads2 for in this material abode1 we are similar to those who enter or depart from a foreign region1 not only in casting aside our native manners and customs1 but from the long use of a strange country1 we are imbued with affections1 manners1 and laws foreign from our natural and true region1 and with a strong propensity to these unnatural habits4 ,such an one1 therefore1 should not only think earnestly of the way1 however long and laborious1 by which he may return to his own1 but that he may meet with a more favourable reception from his proper kindred1 should also meditate by what means he may divest himself of every thing alien from his true country1 which he has contracted2 and in what manner he may best recal to his memory1 those habits and dispositions without which he cannot be admitted by his own1 and which1 from long disuse1 have departed from his soul4 ,in like manner1 it is requisite1 if we wish to return to such things as are truly our own1 and proper to man considered as a rational soul1 to lay aside whatever we have associated to ourselves from a mortal nature1 together with all that propensity to material connections1 by which the soul is allured1 and descends into the obscure regions of sense2 but to be mindful of that blessed and eternal essence intellect1 our true father1 and hastening our return to the contemplation of the uncoloured light of good1 to take especial care of these two things2 one1 that we divest ourselves 7as of foreign garments7 of every thing mortal and material2 the other how we may return with safety1 since thus1 ascending to our native land1 we are different from ourselves before we descended into mortality4 ,for we were formerly intellectual natures2 and even now we are essences purified from every stain contracted by sense1 and from that part which is destitute of reason3 but we are complicated with sensible connections1 on account of our impotence and infirmity1 which is the cause that we cannot always be conversant with intellectual concerns2 but with mundane affairs we can be present with frequency and ease2 for all our energetic powers are stupified and clouded with oblivion1 through body and sense2 the soul not remaining in an intellectual state2 7as the earth when badly affected1 though good fruit is deposited in its bosom1 produces nothing but weeds72 and this1 through the improbity of the soul1 which does not1 indeed1 destroy its essence1 while it acquires brutality2 but by such an accession it becomes complicated with a perishing nature1 is bound in the dark folds of matter1 and is drawn aside from its proper state1 into one that is foreign and base4 ,so that it is highly requisite to study1 if we are solicitous of returning to our pristine state of felicity1 how to depart from sense and imagination1 and her attendant brutality1 and from those passions which are raised by her phantastic eye1 as much as the necessity of our nature will permit4 ,for the intellect must be accurately composed2 and it is proper it should obtain a peace and tranquility free from the contentions of that part which is destitute of reason1 that we may not only hear with attention concerning intellect and intelligible objects1 but to the utmost of our ability1 may enjoy their contemplation2 and thus1 being reduced into an incorporeal nature1 may truly lead an intellectual life1 and not in a false delusive manner1 like these who are at the same time entangled with corporeal concerns4 ,we must1 therefore1 divest ourselves of the various garments of mortality by which our vigour is impeded2 as well this visible and fleshly garment1 as that more interior one with which we are invested contiguous to the skin4 ,we must enter the place of contest naked1 and without the incumbrance of dress1 striving for the most glorious of all prizes1 the ,olympiad of the soul4 ,but the first requisite1 and without which it is not lawful to contend1 is1 that we strip off our garments4 ,and since our vestments are some of them exterior1 and some interior1 so with respect to the denudation of the soul1 one process is by things more open1 another by such as are more occult4 ,for instance1 not to eat1 or not to accept what is offered1 is among things obvious and open2 but not to desire is more obscure2 so that it is here requisite not only to abstain from things improper in deeds1 but likewise in desire4 ,for what does it profit to abstain in actions from what is base1 in the mean time adhering to the causes which produce such actions1 as if bound in indissoluble chains8 ,but this receding from material affections is brought about partly by force1 and partly by persuasion2 and by the assistance of reason the affections languish1 and are1 as it were1 buried in oblivion1 or in a certain philosophical death2 which is1 indeed1 the best mode of desertion1 without oppressing the terrene bandage from which the soul departs4 ,for in things which are the objects of sense1 a violent devulsion cannot take place without either a laceration of some part1 or at least a vestige of separation4 ,but vice steals in upon the soul through continual negligence3 and carelessness is produced by not sufficiently attending to intelligible objects2 the affections in the mean time being excited by the drowsy perceptions of sense1 among which must be also reckoned the sensations arising from food4 ,we must therefore abstain1 not less than from other things1 from such food as usually excites the passions of our soul4 ,let us then in this particular enquire a little farther4 ,there are two fountains1 whose noxious streams detain the soul in matter1 and with which1 as if saturated with lethargic potions1 she forgets her own proper speculations3 ,i mean pleasure and grief1 the artificer of which is sense and its perceptions1 together with the operations attendant on the senses1 imaginations1 opinions1 and memory4 ,the passions1 roused by the energies of these1 and the irrational part1 now fattened with noxious nutriment1 draw down the soul1 and avert her inclinations from her native love of true being4 ,it is requisite1 therefore1 that we revolt from these to the utmost of our ability4 ,but true defections can alone take place by avoiding the passions and rash motions produced by the senses4 ,but1 sensation respects whatever moves the sight1 or the hearing1 or the taste1 or the smell4 ,and sense is1 as it were1 the metropolis of that foreign colony of passions which reside in the soul1 and which must be expelled by him who wishes1 while connected with body1 to become an inhabitant of the royal regions of intellect4 ,let us then enquire how much fuel of the passions enters into us through each of the senses2 and this either when we behold the spectacles of horses in the race1 and the labours of the athletic1 or the contests of those who twist and bend their bodies in leaping1 or when we survey beautiful women4 ,for all these insnare us1 unconscious of the danger1 and subject to their dominion the irrational appetite1 by proffered inchantments of every kind4 ,for by all such inchantments the soul1 as if driven into fury1 compels the compound man to leap rashly1 and without reason1 and full of the brutal nature to bellow and exclaim4 ,in the mean time1 the perturbation appearing from without1 being inflamed by the internal1 which was first of all roused by sense4 ,but the vehement motions excited by the hearing1 arise from certain noises and sounds1 from base discourse1 and mixed assemblies2 so that some1 exiled from reason1 behave as if struck mad2 and others1 enervated by effeminate softness1 agitate themselves by a multitude of trifling gesticulations4 ,and who is ignorant how much the soul is fattened1 and infested with material grossness1 by the ointments and perfumes which commend lovers to each other8 ,but why is it necessary to speak of the passions originating from the taste3 in this respect especially1 binding the soul in a double band2 one of which is thickened by the passions excited by the taste2 the other becomes strong and powerful by the different bodies which we receive in food4 ,for as a certain physician observed1 those are not the only poisons which are prepared by the medical art1 but such things as we daily receive for food1 as well liquid as solid1 are to be reckoned among this number2 and much greater danger arises to our life from these1 than to our bodies from poisons4 ,but the touch does all but transmute the soul into body1 and excites in it1 as in a dissonant body1 certain broken and enervated sounds4 ,the remembrance1 imagination1 and cogitation of all these raise a collected swarm of passions1 i4 e4 of fear1 desire1 anger1 love1 emulation1 cares1 and griefs1 they fill the soul with perturbations of this kind1 cloud its intellectual eye with oblivion1 and bury its divine light in material darkness4 ,on which account it is a great undertaking to be purified from all this rout of pollutions2 and to bestow much labour in meditating day and night1 what measures we shall adopt to be freed from these bonds1 and this because we are complicated with sense1 from a certain necessity4 ,from whence1 as much as our ability will permit1 we ought to recede from those places in which we may 7perhaps unwillingly71 meet with this hostile rout2 and it is requisite we should be solicitous not to engage in combat with these dangerous foes1 lest1 through too great a confidence of victory and success1 instead of vigorous contention1 we produce only unskilfulness and indolence40 ,and in the conclusion of the first book1 he adds1 8,for1 indeed1 if it be lawful to speak freely1 and without fear1 we can by no other means obtain the true end of a contemplative1 intellectual life1 but by adhering to the ,deity 7if ,i may be allowed the expression71 as if fastened by a nail1 at the same time being torn away and separated from body and corporeal delights2 having procured safety from our deeds1 and not from the mere attention to words4 ,but if friendship is not to be conciliated with a divinity1 who is only the governor of some particular region1 with any kind of food1 or by the use of animal nutriment1 much less can a gross diet effect an union with that ,god who is exalted above all things1 and who is superior to a nature simply incorporeal2 but after every mode of purgation1 and the greatest chastity of body1 and purity of soul1 we shall scarcely be thought worthy to obtain the vision of his ineffable beauty2 though this is sometimes permitted to him whose soul is well disposed1 and who has passed through life with the greatest sanctity and purity of manners4 ,so that1 by how much the ,father of all exceeds every nature in simplicity1 purity1 and self-sufficiency1 as being infinitely remote from all suspicion of material contagion1 by so much the more ought he who approaches to the ,deity1 to be entirely pure and holy1 first in his body1 and afterwards in the most secret recesses of his soul2 having distributed a purgation adapted to every part1 and being completely invested with purity1 as with a transparent garment1 fit for the intimate reception of divine illumination40 ,thus far ,porphyry1 whose excellent sentiments on this subject are a lasting monument of the elevation and purity of soul which the ,platonic philosophy affords2 and at the same time sufficiently prove the arrogance and ignorance of those who depreciate the wisdom of the ancients1 and consider their greatest philosophers as involved in mental darkness and delusion4 ,but presumption of this kind is continually increased by indolence1 and strengthened by interest2 and it is common to find scribblers of every kind1 laughing at ,plato and his philosophy1 who are too mean for criticism1 and even too insignificant for contempt4 ,let us1 therefore1 leave such in their native inanity1 and listen to the instructions of the divinely elegant ,proclus1 by which we may ascend to the contemplation of true being1 and the ineffable principle of things4 #f4 #ch8,pythagoras and ,plato command us to fly from the multitude1 that we may pursue the most simple truth1 and apply ourselves wholly to the contemplation of real being4 ,from the multitude of exterior people drawing us aside in various ways1 and deceiving us by fallacious appearances4 ,but much more to shun the multitude of interior people2 for this much more distracts and deceives4 ,we must1 therefore1 fly from the various multitude of affections1 the obscure informations of sense1 the shadowy objects of imagination1 and the dusky light of opinion4 ,for every multitude of this kind is so different in itself1 that its parts are contrary to one another2 from whence it is necessary to betake ourselves to the sciences1 in which multitude has no contrariety4 ,for though affections are contrary to affections1 one perception of sense to another1 imaginations to imaginations1 and opinions to opinions1 yet no one science is found contrary to another4 ,in this multitude1 therefore1 of propositions and notions1 we may collect into one the number of sciences binding them in one according bond4 ,for they are so remote from contrariety to each other1 that notion is subservient to notion1 and inferior sciences minister to superior1 depending on them for their origin4 ,above all1 it is here necessary1 from many sciences which pre-suppose one1 to betake ourselves to one science itself1 no longer supposing another1 and in an orderly series to refer them all to this original one4 ,but after science1 and its study1 it will be necessary to lay aside compositions1 divisions1 and multiform discourses1 and from thence to ascend to intellectual life1 to its simple vision1 and intimate perception4 ,for science is not the summit of knowledge1 but beyond it is intellect2 not that intellect only which is separated from soul1 but the illustration infused from thence into the soul1 which ,aristotle affirms to be the intellect by which we acknowledge the principles of science2 and ,timus says1 that this exists in no place but the soul4 ,ascending1 therefore1 to this intellect1 we must contemplate together with it intelligible essence1 by indivisible and simple perceptions1 speculating the simple genera of beings4 ,but after venerable intellect itself1 it will be proper to contemplate that summit of the soul1 by which we are one1 and under whose influence our multitude is united4 ,for as by our intellect we touch the divine intellect2 so by our unity1 and as it were the flower of our essence1 it will be lawful to touch that first one1 from whom all subordinate unities proceed4 ,and by this our one1 we are especially conjoined with divinity4 ,for similitude may be every where comprehended by that which is similar2 the objects of knowledge by science2 things intelligible by intellect2 and the most unifying measures of being1 by the unity of the soul4 ,but this unity and its energy is the summit of our actions2 for by this we become divine1 when1 flying from all multitude1 we retire into the depths of our unity1 and1 being collected into one1 uniformly energize4 ,thus far we admonish to shun the multitude1 by steps proceeding from the order of knowledge3 in the next place1 we shall proceed in the same design by the series of knowable objects4 ,fly then every sensible species1 for they are heaped together1 are divisible1 and perfectly mutable1 and incapable of affording sincere and genuine knowledge4 ,from these dark informations1 therefore1 betake yourself to incorporeal essence2 since every sensible object possesses adventitious unity1 is by itself scattered and confused1 and full of formless infinity4 ,hence its good is divisible1 and adventitious1 distant and separated from itself1 and residing in a foreign seat4 ,when you have ascended thither1 and are placed among incorporeal beings1 you will behold above the fluctuating empire of bodies1 the sublime animal order1 self-moving1 spontaneously energizing in itself1 and from itself possessing its own essence1 yet multiplied1 and anticipating in itself a certain apparition or image of the essence divisible about the unstable order of bodies4 ,you will there perceive many habitudes of reasons1 various proportions1 and according bonds4 ,likewise the whole and parts1 vivid circles1 and a multiform variety of powers2 together with a perfection of souls not-eternal1 not subsisting1 together as a whole1 but1 unfolded by time1 gradually departing from their integrity1 and conversant with continual circulations4 ,for such is the nature of the soul4 ,but after the multitude belonging to souls1 betake yourself to intellect1 and the intellectual kingdoms1 that you may possess the unity of things4 ,there remain in contemplation of a nature ever abiding in eternity1 of life ever flourishing1 intelligence ever vigilant1 to which no perfection of being is wanting1 and which does not desire the chariot of time1 for the full energy of its essence4 ,when you have beheld natures of this exalted kind1 and have seen by how great an interval they are superior to souls2 in the next place enquire whether any multitude is there1 and if intellect1 since it is one1 is also universal2 and again1 since it is uniform1 if not also multiform3 for you will find it subsists after this manner4 ,when1 therefore1 you have intimately beheld this intellectual multitude1 though profoundly indivisible and united1 transport yourself again to another principle1 and having considered1 as in a more exalted rank1 the unities of intellectual essences1 in the last place proceed to unity perfectly separate and free from all things4 ,and when advanced thus far1 lay aside all multitude1 and you will at length arrive at the ineffable fountain of good4 ,and since it appears1 from these various gradations1 that the soul then properly obtains perfection1 when she flies from all external and internal multitude1 and the boundless variety of the universe1 we may likewise conclude from hence1 that our souls do not alone collect their knowledge from the obscure objects of sense1 nor from things particular and divisible discover a perfect whole1 and a perfect one1 but draw forth science from their inmost recesses1 and produce accuracy and perfection from whatever in appearances is inaccurate and imperfect4 ,for it is not proper to suppose that things false and obscure1 should be the principal sources of knowledge to the soul2 and that things discordant among themselves1 which require the reasonings and arguments of the soul1 and which are ambiguous and confused1 should precede science which is immutable2 nor that things variously changed1 should generate reasons abiding in one2 nor that indeterminate beings should exist as the causes of determinate intelligence4 ,it is not1 therefore1 fit to receive the truth of eternal entities from boundless multitude2 nor from sensible objects the judgment of universals2 nor from things destitute of reason1 accurate discrimination of that which is good3 but it is proper that the soul1 retiring into her immortal essence1 should there scrutinize the good and the true1 and the immutable reasons of all things3 for the essence of the soul is full of these1 though they are clouded by oblivion4 ,the soul1 therefore1 beholding exteriors1 enquires after truth1 in the mean time possessing it in the depths of her essence1 and deserting herself1 explores the good in the dark regions of matter4 ,hence1 every one in the pursuit of reality ought to begin with the knowledge of himself4 ,for1 if we constantly extend our views among the multitude of men1 we shall never discern the one species man1 obscured by the multitude1 and distracted by the division and discord1 and the various mutations of those who participate the species4 ,but if we turn our eye inwards1 there1 remote from perturbation1 we shall behold one reason and nature of men2 since multitude is an impediment to the conversion of the soul into herself4 ,for here variety darkens unity1 difference obscures identity1 and dissimilitude clouds similitude2 since species are confused in the folds of matter2 and every where that which is excellent is mixed with the base40 ,thus far ,proclus2 and thus much for our intended ,dissertation4 ,,life ,,and ,,commentaries ,,of ,,proclus4 ,,the ,,life ,,of ,,proclus1 ,,by ,,marinus#ci2 ,,or1 ,,concerning ,,felicity4 ,when ,i consider the magnitude of mind1 and dignity of character belonging to ,proclus1 a philosopher of our time1 and attend to those requisites1 and that power of composition which those ought to possess who undertake a description of his life2 and lastly1 when ,i regard my own poverty of diction1 ,i am inclined to believe it more proper to refrain from such an undertaking1 not to leap over the fosse 7according to the proverb71 and to decline a discourse involved in so much difficulty and danger4 ,but my scruples are something diminished when ,i consider1 on the other hand1 that even in temples1 those who approach to the altars do not all sacrifice alike2 but that some are solicitously employed in preparing bulls1 goats1 and other things of a similar kind1 as not unworthy the beneficence of the ,gods to whom those altars belong3 likewise that they compose hymns1 some of which are more elegant in verse1 but others in prose2 while some1 who are destitute of all such gifts1 and sacrifice with nothing more perhaps than a cake and a small quantity of bread1 with frankincense1 and who finish their invocations with a short address to the particular divinity they adore1 are not less heard than others4 ,while ,i thus think with myself1 ,i am afraid1 according to ,ibycus#dj1 lest ,i should not offend against the ,gods 7for these are his words7 but against a wise man1 and thus obtain the praise of men4 ,for ,i do not think it lawful1 that ,i who was one of his familiars1 should be silent concerning his life2 and should not1 according to my utmost ability1 relate such particulars concerning him as are true1 and which perhaps ought to be published in preference to others4 ,and indeed by such a neglect ,i shall not perhaps obtain the esteem and honour of mankind1 who will not entirely ascribe my conduct to the desire of avoiding ostentation1 but will suppose ,i avoided such a design from indolence1 or some1 more dreadful disease of the soul4 ,incited1 therefore1 by all these considerations1 ,i have taken upon me to relate some illustrious particulars of this philosopher1 since they are almost infinite1 and may be depended on for their undoubted reality4 ,i shall begin therefore not according to the usual manner of writers1 who are accustomed to distribute their discourse into chapters2 but ,i consider that the felicity of this blessed man ought1 with the greatest propriety1 to be placed as the foundation of this treatise4 ,for ,i regard him as the most happy of those men who were celebrated in former ages2 ,i do not say happy only from the felicity of wisdom1 though he possessed this in the highest degree of all men2 nor because he abundantly enjoyed the goods of an animal life2 nor again on account of his fortune1 though this belonged to him in a most eminent degree1 for he was supplied with a great abundance of all such things as are called external goods3 but ,i call him happy1 because his felicity was perfect1 complete in all parts1 and composed from each of the preceding particulars4 ,having then in the first place distributed#da virtues according to their kinds into natural1 moral1 and political1 and also into those of a sublimer rank1 which are wholly conversant with purification and contemplation1 and are therefore called ,cathartic and ,theoretic1 and also such as are denominated ,theurgic1 by which we acquire a similitude with some particular divinity2 but omitting such as are superior to these1 as beyond the reach of man1 we shall begin from such as are more natural1 and which are first in the progressions of the human soul1 though not first in the nature of things4 ,this blessed man1 then1 whose praise is the subject of this treatise1 naturally possessed1 from the hour of his birth1 all those physical virtues which fall to the lot of mankind2 the traces of which were manifest in the latest period of his life1 and appeared to surround and invest his body after the manner of a tenacious shell4 ,in the first place1 he was endued with a singular perfection of sensation1 which they denominate corporeal prudence2 and this was particularly evident in the nobler senses of seeing and hearing1 which are indeed given by the gods to men for the purpose of philosophizing1 and for the greater convenience of the animal life2 and which remained entire to this divine man through the whole of his life4 ,secondly1 he possessed a strength of body which was not affected by cold1 and which was neither weakened nor disturbed by any vicious or negligent diet1 nor by any endurance of labours1 though it was exhausted day and night1 while he was employed in prayer1 in perusing the works of others1 in writing books himself1 and in conversing with his familiars2 all which he performed with such expedition1 that he appeared to study but one thing alone4 ,but a power of this kind may with propriety be called fortitude of body1 from the singular strength employed in such exertions4 ,the third corporeal virtue with which he was endued was beauty1 which1 when compared with temperance1 the authors of these appellations have very properly considered as possessing a similitude of nature4 ,for as we consider temperance as consisting in a certain symphony and consent of the powers of the soul1 so corporeal beauty is understood to consist in a certain agreement of the organical parts4 ,he was indeed of a most pleasing aspect1 not only because he was endued with this excellent proportion of body1 but because the flourishing condition of his soul beamed through his corporeal frame like a living light1 with splendors too wonderful for language to explain4 ,and indeed he was so beautiful that no painter could accurately describe his resemblance2 and all the pictures of him which were circulated1 although very beautiful1 were far short of the true beauty of the original4 ,but the fourth corporeal virtue which he possessed was health1 which they affirm corresponds to justice in the soul2 and that this is a certain justice in the disposition of the corporeal parts1 as the other in those of the soul4 ,for justice is nothing more than a certain habit1 containing the parts of the soul in their proper duty4 ,hence1 that is called health by physicians1 which conciliates the jarring elements of the body into union and consent2 and which ,proclus possessed in such perfection1 that he affirmed he was not ill above twice or thrice1 in the course of so long a life as seventy-five years4 ,but a sufficient proof of this is evident from hence1 that1 in his last illness1 he was entirely ignorant what the disorders were which invaded his body1 on account of the great rarity of their incursions4 ,such then were the corporeal goods which ,proclus possessed1 and which may be called the forerunners1 and as it were messengers1 of those forms into which we have distributed perfect virtue4 ,but the first powers and progeny of his soul1 which he naturally possessed1 previous to instruction1 and those parts of virtue with which he was adorned1 and which ,plato reckons the elements of a philosophic nature#db1 must excite the wonder of any one who considers their excellent quality4 ,for he was remarkable for his memory and ingenuity2 he was of a disposition magnificent1 gentle1 and friendly2 and a companion1 as it were1 of truth1 justice1 fortitude and temperance2 and his love of truth was so great1 that he never admitted any prudent dissimulation1 but hated falsehood vehemently4 ,indeed it is necessary that he who prosecutes truth1 with so much earnestness and sincerity1 should be extremely desirous of it from his infancy1 since truth is the source of every good1 both to gods and men4 ,but that he despised corporeal pleasures1 and was an eminent lover of temperance1 is sufficiently evident from his great propensity to disciplines1 and his desire of every kind of studies2 for dispositions of this kind never suffer beastly and illiberal pleasure to dwell it the mind1 but are able to excite in the soul1 from her own internal operations1 sincere pleasure and delight4 ,but it is impossible to say how foreign he was from avarice1 so that when a boy he despised the wealth of his parents1 though very rich1 on account of his incredible love towards philosophy4 ,hence he was far removed from illiberality1 and from the care of lesser concerns1 as he was most studious of the universe1 and of every thing divine and human4 ,but from such a disposition of the rational soul1 having acquired true magnanimity1 he considered human life as of no account1 and1 unlike the multitude1 viewed nothing dreadful in death4 ,so that he by no means feared all that rout of molestations which appear terrible to others1 and this in consequence of that natural affection which it is proper to call by no other name than that of ,fortitude alone4 ,but1 from all these virtues1 ,i think it must be evident to those who have not experienced his best of dispositions1 that he loved equity from a boy2 that he was just and mild1 and by no means difficult or unjust in his associations or contracts4 ,to us indeed he certainly appeared modest and elegant1 neither avaritious nor illiberal1 neither arrogant nor timid4 ,but will it not be superfluous to mention the goodness and fertility of his ingenuity8 ,especially among those who know and who have heard1 that he was full of the most beautiful disciplines1 and who are acquainted with the multitude he produced and published to the world1 so that he alone seemed to have drank nothing of the cup of oblivion1 as he was endued with a power of memory which was never disturbed1 and that which belongs to the oblivious1 never happened to him4 ,besides1 he never neglected fresh acquisitions1 as if possessing a sufficiency of disciplines2 and as one who is merely delighted with their study4 ,but he was most remote from a nature rustic and horrid1 and averse from the ,muses1 and particularly propense to more cultivated endowments3 for on account of his singular urbanity and festivity 7without transgressing the bounds of true honesty7 in his common associations1 sacred feasts1 and other actions1 he allured and charmed his familiars1 and always dismissed them more cheerful and pleased4 ,his mother1 therefore1 ,marcella1 lawfully united to his father ,patricius1 both of the ,lycian nation1 and excelling in birth and virtue1 produced our philosopher1 thus endued from the beginning with all these1 and other gifts of nature4 ,and#dc ,minerva1 the tutelar goddess of ,byzantium1 received him when born1 and took care of him as a midwife1 she being the cause of his birth in that city3 but afterwards she provided for his well-being1 when he was numbered among boys and young men2 for she appeared once to him in a dream1 and exhorted him to the study of philosophy1 from whence arose his great propensity to this goddess1 as he particularly performed her sacred rites1 and cultivated with a greater fury 7as ,i may say7 her institutions4 ,lastly1 his parents brought him1 when born1 into their native country ,zanthus1 consecrated to ,apollo3 and ,i cannot but think that this country happened to him by a certain divine providence2 as it was requisite that he1 who was to be the prince of all sciences1 should be educated under the presiding deity of the ,muses4 ,here1 being instituted in the most elegant manners1 he pursued moral virtues1 and was accustomed to right conduct1 and to a declination of its contrary1 that which is base4 ,but at that time the love of the gods1 who had attended him from his nativity1 manifestly appeared2 for being once detained by some disease of body1 and it appearing very difficult1 and scarcely possible to cure him1 there stood at his bed a youth of a more than ordinary appearance1 so that even previous to the declaration of his name1 he might be considered as#dd ,telesphorus or ,apollo3 but the god proclaiming who he was1 and pronouncing his name1 touched the head of ,proclus 7for he stood reclining his head on ,proclus' pillow7 and having immediately restored him to health1 vanished from his sight4 ,and such was the#de divine vision1 and the divine benevolence at that time exhibited to our youth4 ,but having1 for a short space of time1 in ,lycia1 applied himself to grammar1 he went to ,alexandria in ,egypt1 bringing with him very singular moral virtues1 by which he excited towards himself the love of the masters resident in that place4 ,hence ,leonas the rhetorician1 who derived 7as ,i think7 his lineage from ,isaurus1 and was illustrious among many of that profession1 who were then at ,alexandria1 not only made him a partaker of his studies1 but thought him worthy to become his domestic1 and ordered that he should be supplied with food1 together with his wife and children1 no otherwise than if he had been his#df true son4 ,he likewise took care to procure him the notice of the principal men in ,egypt1 who being wonderfully delighted with the acumen of the youth's ingenuity1 and with the elegance and integrity of his morals1 reckoned him among their greatest friends4 ,but he was also instructed by ,orion the grammarian1 whose ancestors discharged the sacerdotal office among the ,egyptians1 and who had made such a progress in the knowledge of his art1 that he composed elaborate books on this subject1 which he left not without advantage to posterity4 ,he also went to the schools of the ,roman preceptors1 and made a great progress in that language2 for he was at first led to the study of his father's profession1 in which he was illustrious1 his employment being the study of law in the royal city4 ,but when it appeared how vehemently the young man was delighted with the study of rhetoric1 as he had not yet touched the writings of the philosophers1 he both acquired great glory from his acquisitions1 and became the admiration of his fellow pupils and masters1 on account of the elegance of his discourse1 and his celerity in perceiving2 and from his exhibiting more the habit and industry of the master1 than that of the scholar4 ,but while he yet frequented the rhetorical school1 the sophist ,leonas1 made him the companion of his journey to ,byzantium3 which he undertook for the purpose of gratifying his friend ,theodorus1 who was at that time prfect of ,alexandria1 and who was a man both polite and magnificent1 and a lover of philosophy4 ,but ,proclus1 though a youth1 followed his master more cheerfully in this journey1 lest he should be compelled to interrupt his studies4 ,however1 that ,i may speak more truly1 a certain good fortune brought him back to the source of his nativity4 ,for1 on his return1 his tutelar goddess exhorted him to philosophy1 and to visit the ,athenian schools4 ,but having first returned to ,alexandria1 and bid farewel to rhetoric1 and the other arts which he had formerly studied1 he gave himself up to the discourses of the philosophers then resident at ,alexandria4 ,but he frequented#dg ,olympiodorus1 the most illustrious of philosophers1 for the sake of imbibing the doctrine of ,aristotle2 and in mathematical disciplines gave himself to ,hero#dh1 a religious man1 and one who was eminently skilful in the proper methods of institution4 ,but these men were so delighted with the manners of the youth1 that ,olympiodorus wished him to espouse his daughter1 whom he had taken care to instruct in philosophy1 and ,hero committed to him all his religion1 and made him his constant companion4 ,but having1 on a certain time1 heard ,olympiodorus1 a man who was endued with a great power of speaking1 and on account of the celerity of his speech1 and the gravity of his subjects1 was understood by very few of his auditors1 as he was departing with the dismissed multitude1 he repeated to his companions all that was said1 and almost verbatim1 though the discourse was copious2 as ,ulpianus ,gazus1 one of his fellow-disciples informed me1 who also consumed not the least part of his life in the study of philosophy4 ,but he likewise learned1 with great facility1 the writings of ,aristotle1 pertaining to rational philosophy#di1 though the bare reading of them is difficult to those who are engaged in the attempt4 ,having therefore1 at ,alexandria1 applied himself to these masters1 and enjoyed their confidence in such instruction as they were able to afford1 when upon reading together with them a certain author1 they appeared to him not to interpret the mind of the philosopher as they ought2 conceiving a contempt for these schools1 and at the same time being mindful of the exhortation which had been divinely sent to him at ,byzantium1 he went to ,athens1 attended by the presiding deities of eloquence and philosophy1 and by beneficent dmons4 ,for that he might preserve the genuine and entire succession#ej of ,plato1 he was brought by the gods to the guardian city of philosophy1 as the circumstances which happened on his first entrance into the city1 and all the divine excitations manifestly evince3 for they openly presaged1 that this gift was sent from the father ,apollo1 and was a future suffrage of his succession confirmed by divine events4 ,for when his vessel drove to the ,pyrum1 and it was told to the citizens1 ,nicolaus1 who afterwards flourished in the rhetorical art1 but at that time studied under the masters of ,athens1 descended to the shore as if to an acquaintance1 received him for his guest as if he had been a citizen1 and brought him to the city2 for ,nicolaus was also a ,lycian4 ,but ,proclus1 who perceived himself weary from his journey1 sat down at the temple of ,socrates1 though he did not yet know1 nor had heard that ,socrates was worshipped in that place1 and requested ,nicolaus that he would stay there for a short time1 and1 if possible1 procure him some water1 as he said that he was exceeding thirsty4 ,immediately ,nicolaus1 from that very consecrated place1 brought him some water2 for the fountain belonging to the statue of ,socrates was not far distant4 ,but while he was drinking1 ,nicolaus1 for the first time1 considering the circumstance3 ,this is an omen1 says he1 because you have sate in the temple of ,socrates1 and have there first drank the ,attic water4 ,then ,proclus rising1 and having paid due reverence to the place1 proceeded to the city4 ,but when he came to the tower1 the porter who was present at his entrance1 and was about to close the gates with bars1 said to him1 7that ,i may repeat the words of the man17 8,certainly unless you had come1 ,i should have closed the gates40 ,and what omen could be more manifest1 or could less require the interpretations of#ea ,polletes or ,melampodes1 or such like diviners8 ,but ,proclus1 despising the schools of the rhetoricians1 though they very much desired his association1 as if he had come for that very purpose1 met with the prince of philosophers ,syrianus#eb1 the son of ,philoxenus4 ,at that time too1 ,lachares was in the same company1 a man much conversant in philosophy1 and the companion of ,syrianus in his study1 but in eloquence he was in as great admiration as ,homer in the poetic art4 ,he then was1 as ,i have said1 present at the same time4 ,but it was now the evening twilight1 and while they were engaged in mutual converse1 the sun sate1 and the moon made her first appearance after her change3 wherefore1 having saluted the stranger1 they endeavoured to dismiss him1 as being a young man1 from their company1 that they might adore the goddess apart4 ,but he not having proceeded far1 beheld also the moon appearing from the same house1 and laying aside his sandals1 in their presence saluted the goddess4 ,here ,lachares1 admiring the confidence of the youth1 said1 speaking to ,syrianus1 8,this is what ,plato#ec divinely affirms of great geniuses2 that they either produce great good1 or its contrary40 ,and such1 that ,i may mention a few out of many1 were the signatures of divine original1 which happened to our philosopher1 on his first arrival at ,athens4 ,but ,syrianus brought him to the great ,plutarch#ed1 the son of ,nestor1 who1 when he saw the young man1 not yet twenty years old1 and heard of his love and desire of a philosophic life1 he was much delighted1 and immediately made him a partaker of his philosophic study1 though his age almost forbade such an attempt2 for he was then very old4 ,he therefore read to ,proclus his commentary on ,aristotle's books on the soul1 and on the ,phdo of ,plato3 and this great man exhorted him to commit to writing what he heard1 employing the ambition of the youth as an instrument1 by telling him1 that if he completed those scholia1 they would be reported as the commentaries of ,proclus4 ,and as he loved the youth very much on account of his inclination to the best studies1 he called him his son1 and caused him to reside with him as a domestic4 ,but after he saw his temperance1 with regard to animal food1 he exhorted him not to abstain from animals entirely1 but to use them so far as was necessary to the vital energies of the corporeal part4 ,he likewise gave the same advice to ,syrianus1 concerning the youth's diet4 ,but he answered the old man1 as that divine head 7,proclus7 informed us3 8,suffer him with that frugality to learn what ,i wish1 and then1 if he pleases1 he may die40 ,such was the care of his masters respecting him1 in every concern4 ,but the old man lived about two years after the arrival of ,proclus2 and dying1 commended the youth to his successor ,syrianus1 as also his grandson ,archiadas4 ,but ,syrianus1 when he had received ,proclus as his pupil1 not only much assisted him in learning1 but made him his domestic as to other concerns1 and the companion of his philosophic life1 having found him such an auditor and successor as he had for a long time sought for2 and one who was adapted for the reception of a multitude of disciplines1 and divine dogmata4 ,in a shorter space1 therefore1 than two years1 he read1 together with ,syrianus1 all the works of ,aristotle1 i4 e4 his logic1 ethics1 politics1 physics1 and theological science4 ,but being sufficiently instructed in these1 as in certain#ee proteleia1 and small mysteries1 ,syrianus led him to the sacred discipline of ,plato1 and this by an orderly progression1 and not#ef according to the oracle1 with a transcendent foot4 ,and he was careful that he might survey with him true mysteries1 with the eyes of his soul1 free from material darkness1 and with a speculation of intellect refined and pure4 ,hence ,proclus was employed night and day in vigilant energies1 and in writing compendiously what he had heard1 employing his own judgment in the selection and order4 ,and in consequence of this unwearied assiduity1 he made so great a progress in a short time1 that by then he was twenty-eight years of age1 he composed a multitude of works2 and among the rest his very learned and elegant commentaries on the ,timus4 ,but from an institution of this kind1 his manners also received a greater ornament1 since as he advanced in science he accumulated virtue4 ,in the original 1 which ,i wonder ,fabricius should translate1 quod aiunt1 as it is usual with the ,platonists1 to cite the ,zoroastrian oracles exactly in these words1 instances of which may be found in ,proclus on ,plato's theology2 and the very words prove themselves to be a part of an oracle1 when attentively considered4 ,but he likewise pursued the civil virtues from ,aristotle's political writings1 and ,plato's books concerning laws and a republic4 ,however1 lest he should be conversant only in the contemplation of these1 and should attempt nothing actual1 since he was hindered from engaging in public affairs himself1 from his being employed in more important concerns1 he exhorted ,archiadas1 who was a religious man1 to a political life1 at the same time instructing him in its nature and virtues1 and explaining to him the methods proper to be observed in the discharge of its duties4 ,and in the same manner as they incite those who run in the race1 so he exhorted him to preside in common over the whole of his city1 and to confer benefits privately upon each according to all the virtues1 but particularly in conformity with the laws of justice4 ,but he excited in reality the emulation of himself in ,archiadas1 when he exhibited to him his own liberality and magnificence with respect to riches1 by bestowing gifts at one time on his friends1 and at another time on his kindred1 guests1 and citizens2 proving himself by this means not in the least affected with avarice1 and conferring not a little to the good of public concerns4 ,but when dying1 he bequeathed that part of his possessions which he did not leave to ,archiadas1 to his own country1 and to ,athens4 ,but ,archiadas became so great a lover of truth1 as well from the company of his own associates as from the friendship of ,proclus1 that he is never mentioned by our acquaintance1 without being at the same time celebrated as the religious ,archiadas4 ,but the philosopher also employed himself in civil consultations among public assemblies1 where things pertaining to the republic were discussed2 giving the most prudent counsels1 conferring with governors concerning equity1 and not only exhorting them to an impartial distribution of justice1 but in a manner compelling them by philosophical authority4 ,for he had a certain public care of the morals of princes1 and not only instructed them in the art of temperate government by his discourse1 but also by his own example through the whole of his life2 since he was1 as it were1 the exemplar of temperance to the rest4 ,but he gave a specimen of civil fortitude perfectly ,herculean3 for since at that time there was1 as it were1 a sea of troubles upon him1 and mighty waves of stormy employments were roused by adverse winds against his upright life1 he conducted himself1 though in danger1 with gravity1 and an unshaken constancy4 ,and when he was once very much molested by the improbity of some violent men1 which was both pernicious and dangerous to himself1 he undertook a journey into ,asia1 which contributed greatly to his own advantage3 for as he was not unskilful in the more ancient rites of that place1 which he yet preserved1 a divine power afforded him this occasion of departure4 ,hence1 as he well knew the whole of their concerns1 he taught them more accurately in things pertaining to the gods1 if they happened to have neglected any thing through a long interval of time4 ,and while he was engaged in all these employments1 and lived in a correspondent manner1 he was so concealed from the multitude that he even excelled the ,pythagoreans1 who keep with unshaken constancy this precept of their founder1 1 live concealed4 ,but having passed a year only in the parts of ,lydia1 he returned to ,athens1 under the providential protection of the presiding goddess of philosophy4 ,and thus fortitude was perfected in our philosopher1 first by nature1 then by custom1 and afterwards by science and the consideration of causes4 ,besides this1 he exhibited in another manner his politic habit practically1 by composing letters for noblemen2 and by this means procuring good to entire cities4 ,but of this ,i have a sufficient testimony from those on whom they were bestowed1 as well ,athenians as ,argives1 and others of different nations4 ,but he likewise much promoted and increased literary studies1 demanding of princes rewards for the preceptors1 according to their several deserts4 ,nor did he undertake this rashly1 nor with any interested views1 but he compelled them 7as he considered it a matter of great moment7 to be diligent in their profession1 interrogating and discoursing with them respecting every particular3 for he was a judge sufficiently instructed in the employments of them all4 ,and if he ever found any one negligent in his profession1 he sharply reproved him2 so that he appeared very vehement and ambitious1 because he was both willing and able to give a just determination on every subject3 and he was indeed a lover of glory4 ,but this was not a fault in him1 as in most1 because it alone regarded virtue and goodness4 ,and1 perhaps1 without an energy of this kind1 nothing great and excellent would ever subsist in the human mind4 ,but he was in this respect vehement3 this ,i will not deny4 ,yet1 at the same time1 he was gentle2 for he was easily pleased1 and demonstrated in a moment that his anger was as pliable as wax4 ,for1 almost at the same time1 he was 7as ,i may say7 wholly transported in reprehension1 and with a desire of becoming subservient to their interest1 and that he might intercede with princes in their names2 being moved with a certain natural conjunction of soul1 and1 as it were1 sympathy of grief4 ,and here1 ,i very opportunely recollect a peculiar example of his natural sympathy of soul with others3 nor do ,i think the like was ever related of any other man4 ,for1 notwithstanding he was unmarried1 and had no children 7because he was not desirous of such connections1 but remained free from them all1 though many noble and rich alliances were offered him71 yet his care of all his familiars and friends1 and of their wives and children1 was as great as if he had been some common father1 and the cause of their birth2 for he bestowed a singular attention on the life of each4 ,and whenever any one of them was detained by any disease1 he first earnestly supplicated the gods on their behalf with sacrifices and hymns2 afterwards he gave a prompt attendance on the sick person himself1 convened the physicians1 and urged them to make an immediate trial1 if they knew of any thing in their art advantageous to the condition of the diseased2 and sometimes he produced some singular advice himself1 among the physicians2 and thus delivered many from imminent dangers4 ,and the greatness of this blessed man's humanity towards his servants1 may be understood by those who desire it1 from his will4 ,but of all his familiars1 he loved ,archiadas and his kindred the most2 because1 in the first place1 their succession was derived from the genus of ,plutarch the philosopher2 and afterwards on account of that#eg ,pythagoric friendship which he maintained with ,archiadas1 as he was both the companion and preceptor of his studies4 ,and this other kind of friendship1 differing from the two already mentioned1 appears to have been the most firm and excellent1 among these illustrious men4 ,for ,archiadas desired nothing1 which was not also the wish of ,proclus2 and on the contrary1 the desires of ,proclus were the constant wishes of ,archiadas4 ,but having now brought the political virtues1 which are inferior to the true ones1 to an end1 and terminating them in#eh friendship1 as their proper bound1 we shall now pass to the ,cathartic differing from the politic virtues4 ,indeed1 the employment of these last consists in purifying the soul1 that so being liberated from the body as much as they are able to effect1 it may regard human concerns1 and possess a certain similitude with divinity2 which is the soul's best and most exalted end4 ,yet they do not all liberate after the same manner1 but some more1 and others less4 ,since there are certain political purgations which adorn their possessors1 even while connected with body1 and reduce them to a better condition2 bringing under the dominion of reason1 anger and desire1 and entirely destroying passion and every false opinion3 but the ,cathartic virtues1 which are superior to these1 separate entirely from this truly leaden weight of body1 and procure an easy flight from mundane concerns4 ,and in these1 indeed1 our philosopher was studiously employed during the whole of his life1 which was devoted to philosophy2 since he both taught by his discourses what they were1 and after what manner they were preparatory to felicity1 and in a particular manner conformed his life to their institutions2 performing every thing which could contribute to the separation of his soul1 using both night and day prayers1 lustrations1 and other purifications1 as well according to the ,orphic as the ,chaldaic institutions3 and every month he descended1 with great diligence1 to the sea2 and this sometimes twice or thrice4 ,but he was exercised in these1 not only in the vigour of his age1 but also towards the close of his life2 and these customs he observed perpetually1 as if they were certain invariable statutes4 ,but he used meat and drink1 and other necessary pleasures1 only so far as was necessary to avoid the molestations of disease2 for he was in these by much the most frugal1 and particularly loved abstinence from animal food4 ,and if at any time he was invited to eat it more vehemently1 he was so cautious in its use1 that he ate it merely after the manner of a taster4 ,but he purified himself every month by the sacred rites1 in honour of the mother of the gods1 celebrated by the ,romans1 and prior to them by the ,phrygians3 he likewise more diligently observed the unfortunate days of the ,egyptians than they themselves2 and1 besides this1 fasted1 on certain days in a peculiar manner1 on account of the lunar appearances#ei4 ,he likewise instituted a fast on the last day of the month1 not having supped the day before4 ,but in what a splendid manner1 and with what piety1 he celebrated the new moon1 and properly observed1 with sacrifices1 the more illustrious feasts of almost all nations1 according to the manner of each country2 and how from these he did not1 according to the custom of many1 take occasion of becoming idle and intemperate1 but employed himself in continual prayers1 hymns1 and the like1 his hymns sufficiently evince1 which not only celebrate the divinities of the ,greeks1 but likewise ,marna ,gazus1 ,esculapius ,leonteuchus1 ,ascalonites1 and ,theandrites1 another god much venerated by the ,arabians2 together with ,isis1 worshipped by the ,philians2 and lastly1 all the rest which were the subjects of his devotion4 ,for this sentiment was very familiar to this most religious man1 that it was proper a philosopher should not be careful in the observance of the rites and institutions of one particular city1 nor of certain nations only1 but that he should be the general priest of the universe4 ,and thus was he pure and holy1 so far as pertains to the virtue of temperance4 ,but he declined1 as much as possible1 pain3 and if it ever happened to him1 he bore it with gentleness1 and diminished it with this view1 that his best part might not at the same time be affected with its molestations4 ,and the fortitude of his soul in this respect1 was sufficiently evinced in his last illness2 for when1 at that time1 he was oppressed and tormented with the most excruciating pains1 he endeavoured to the utmost to mitigate and expel their afflictive invasions4 ,hence1 on such occasions1 he often commanded us to repeat certain hymns1 which when repeated procured him a remission and cessation of pain4 ,and what is more wonderful1 he remembered what he heard of these1 though forgetful of almost all human concerns1 from the dissolution of his corporeal part continually increasing4 ,for when we began to repeat1 he supplied what was unfinished of the hymns1 together with many of the ,orphic verses2 for it was these we were then reciting4 ,nor was he only thus constant in enduring corporeal evils1 but much more so in external unfortunate events1 and such as appeared to happen contrary to expectation4 ,so that he would say1 concerning particulars of this kind1 ,so it is1 such things are usual2 which seemed to me1 or rather at that time appeared to be1 worthy of remembrance1 and an evident argument of our philosopher's magnanimity4 ,but besides this1 he restrained anger as much as possible1 so that it might either remain free from all excitation1 or that at least reason might not consent to its indulgence1 but the irrational faculty alone1 contrary to his will1 might be moderately and lightly excited4 ,and with respect to venereal concerns1 he used them in the natural way1 but so as that he might not proceed beyond a very moderate and light phantasy1 in their indulgence4 ,and thus the soul of this blessed man1 having collected itself from all parts1 and retiring into the depths of its essence1 departed after a manner from body2 while it yet appeared to be contained in its dark receptacle4 ,for he possessed a prudence1 not like that of a civil nature1 which is conversant in the administration of fluctuating particulars1 but prudence itself1 by itself sincere1 which is engaged in contemplating1 and converting itself into itself1 without any longer consenting to a corporeal nature4 ,he likewise possessed a temperance free from evil2 and which is not even moderately influenced by perturbations1 but is abstracted from all affections4 ,and lastly1 he acquired a fortitude1 which does not fear a departure from body4 ,but reason and intellect having obtained in him a perfect dominion1 and the inferior powers of his soul no longer opposing themselves to purifying justice1 his whole life was adorned with the divine irradiations of genuine virtue4 ,our philosopher1 therefore1 having most happily absolved this form of virtues1 advancing now1 as it were1 by the highest and most mystical step1 he ascended to the greatest and most consummate or telestic virtues2 employing for this purpose1 the felicity of his nature1 and a sciential institution4 ,hence1 being now purified1 and the victor of his nativity1 and despising the vain ,thyrsus-bearers1 and boasters of wisdom1 he happily penetrated into her profound recesses1 and enjoyed the contemplation of the truly blessed spectacles she contains4 ,no longer requiring prolix dissertations1 or demonstrations1 for the purpose of collecting the science of these1 but1 with a simple vision and energy of intellect1 beholding the exemplar of the divine mind1 he obtained a virtue which cannot with sufficient propriety be called prudence1 but is more properly denominated wisdom1 or something1 if possible1 still more venerable and divine4 ,but the philosopher energizing1 according to this virtue1 easily comprehended all the theology of the ,greeks and ,barbarians1 and whatever is shadowed over by the figments of fables1 and placed it in a clear light1 for the use of those who are willing and able to pursue its latent signification4 ,but having interpreted divinely every thing of this kind1 and shewing the symphony between them all2 at the same time1 investigating all the writings of the ancients1 whatever he found in them of genuine wisdom1 and approved by general consent1 this he judiciously applied to use2 but if he found any thing of a different and dissonant nature1 this he entirely rejected1 as vicious and false4 ,and whatever he met with contrary to wisdom1 though endued with a friendly appearance1 this he vigorously subverted by a diligent examination4 ,nor did he employ less force and perspicuity in his association with other men4 ,for he was a man laborious to a miracle2 as he often1 in one day absolved five1 and sometimes more lectures2 and writ besides1 many verses1 often to the number of seven hundred4 ,besides this1 he went to other philosophers1 and frequented their company2 and again celebrated with them an evening association1 ceasing from the labour of writing4 ,and all these employments he executed in such a manner1 as not to neglect his nocturnal and vigilant piety to the gods1 and assiduously supplicating the sun when rising1 when at his meridian1 and when he sets4 ,but he was the parent of many dogmata1 which were never known before1 both in physics1 and in intellectual and more divine concerns4 ,for he first taught1 that there was a kind of souls#fj1 endued with the power of contemplating many forms at once1 which he placed1 not without great probability1 between intellect1 collectively1 and as it were with one intuition comprehending every thing1 and souls1 which are alone able to direct their vision to one particular form4 ,and those who are willing to peruse his works will meet with a great variety of dogmata1 peculiar to him alone2 the relation of which ,i shall omit1 lest ,i should give a too great extent to my discourse4 ,but he who evolves his writings1 will easily perceive that all we have above related of him is most true1 and much more if he happens to have known him1 to have seen his face1 and to have heard him interpreting in the most excellent dissertations1 and delivering the ,platonic and ,socratic dogmata in his yearly schools4 ,nor did he seem destitute of divine inspiration2 for he produced from his wise mouth1 words similar to the most white and thick falling snow#fa2 so that his eyes emitted a bright radiance1 and the rest of his countenance was resplendent with a divine light4 ,hence1 when on a certain time1 one ,rufinus1 a man of a great name in the republic1 who was studious of truth1 and in other respects worthy of veneration1 came to him when he was teaching and interpreting1 he perceived that the head of ,proclus was surrounded with a light2 and when the philosopher had finished his interpretation1 ,rufinus rising1 adored him2 and offered to give a public testimony1 by oath1 of the divine vision which he had observed4 ,and much gold was offered to him1 by the same ,rufinus1 on his return from ,asia1 having escaped the danger of the war4 ,but ,proclus likewise rejected this gift1 and was by no means willing to receive it4 ,but that we may return to our first design1 having now discoursed concerning the contemplative wisdom of the philosopher1 though in a manner but little suited to its dignity1 it remains that we now speak of the justice pertaining to this kind of virtues4 ,for this1 unlike that of which we spoke before1 is not conversant in distribution1 or proportion2 but must be equally removed from the kind of self-energizing justice1 by which all things are alone directed to the rational soul4 ,for to that1 concerning which we now treat1 it is alone proper to refer every energy to intellect and the deity1 which our philosopher performed in the most exalted manner4 ,for he scarcely rested from his diurnal labours1 or refreshed his body with sleep1 and perhaps even then was not free from meditation and contemplation4 ,this is certain1 that having very speedily roused himself from sleep1 as from a certain torpor of the soul1 he aspired after the morning1 the time of prayer2 and lest the greater part of the night1 should glide from him without advantage1 as he was lying alone in his bed1 he either composed hymns1 or examined and fortified those dogmata which afterwards1 in the day time1 he committed to writing4 ,after a similar manner he pursued that temperance which has an affinity with these virtues1 and which consists in a conversion of the soul to intellect1 so as not to suffer itself to be touched1 nor moved with any other concerns4 ,lastly1 he joined fortitude in alliance with these1 by a certain perfect method1 zealously aspiring after that liberty which is ignorant of all passion1 and which he perceived was natural to the divine object of his contemplation4 ,and thus1 through the whole of his conduct1 he did not lead the life of a man merely good1 to which1 as ,plotinus says1 the political virtues may lead1 but leaving this far behind him1 he endeavoured to change it for one much more perfect and divine1 the life of the gods themselves2 since1 to become similar to these1 and not to virtuous men1 was the great object of his contention4 ,and thus he had rendered virtues of this kind familiar to himself1 while he frequented the philosopher ,syrianus1 and evolved and studied the commentaries of the ancients4 ,but he received from the mouth of his preceptor certain small seeds1 as it were1 of the ,orphic and ,chaldaic theology2 because he was prevented from hearing the complete interpretation of his master on the ,orphic verses4 ,for ,syrianus left to the choice of ,proclus and one ,domninus#fb1 a philosopher of the ,syrian nation1 and who afterwards succeeded ,syrianus1 the exposition of the ,orphic writings1 or the oracles4 ,but they were by no means unanimous in their choice2 for ,domninus preferred the interpretation of the ,orphic verses1 and ,proclus that of the oracles4 ,but our philosopher did not perfect his undertaking1 because the death of the great ,syrianus happened not long after4 ,having therefore1 as ,i have said1 received the outlines from the mouth of his master1 he applied himself with the greatest diligence to the written commentaries of ,syrianus upon ,orpheus2 and being assiduously nourished with the copious lucubrations of ,porphyry and ,jamblichus on the oracles1 and similar writings of the ,chaldeans1 he arrived1 as much as is possible to man1 to the top of those highest virtues1 which the divine ,jamblichus was accustomed to call after a truly divine manner1 theurgic4 ,he laboured therefore1 not without exquisite judgment1 in collecting the expositions of philosophers prior to his time2 and contracted into one1 other ,chaldaic hypotheses1 and the most excellent of the commentaries on the divine oracles1 completing this great work in the space of five years2 concerning which this divine vision appeared to him in his sleep4 ,for he saw the great ,plutarch approach to him1 affirming that he should live so many years as he had composed tetrads or quaternions on the oracles4 ,afterwards1 having collected the number of these1 he found they amounted to seventy4 ,but that this was a divine dream1 was sufficiently evinced by the last part of his life2 for though he lived seventy-five years1 as we have mentioned above1 yet he had not the perfect use of his powers1 in the five last4 ,but his body1 though naturally of the best constitution1 being debilitated by that hard and scarcely tolerable kind of diet which he used1 and by so many frequent labours and fastings1 began to languish exactly at his seventieth year1 so that he then became much more remiss in all his energies than before4 ,yet1 even at this period1 and thus affected1 he composed orations and hymns3 he also writ some things1 and conversed with his friends2 but his ancient vigour was wanting in each4 ,hence the memory of the dream excited his wonder1 and he every where said1 that he had only lived seventy years4 ,but while he laboured under this infirmity of body1 a certain youth1 named ,hegias1 rendered him more alert in the business of interpretation4 ,this young man1 who already exhibited from his tender years such egregious signs of all the virtues of his ancestors1 was one of that golden chain of philosophers1 who had formerly appeared to men2 and adhered most diligently to ,proclus delivering the ,platonic and other theologies4 ,but ,proclus1 at this advanced period1 was not moderately rejoiced in communicating with the young man1 his own writings1 after he understood that he made cubital advances in every kind of disciplines4 ,and thus we have briefly discoursed concerning the apparatus of the philosopher in the ,chaldean oracles4 ,but ,i1 on a certain time1 having read with him the ,orphic verses1 and heard1 among his interpretations1 not only the recondite theology which is to be found in ,jamblichus and ,syrianus1 but also1 among many other divine men1 ,i requested the philosopher that he would not leave these divine verses also without his explanation3 but his answer was1 ,that he had often thought of writing commentaries on ,orpheus1 but that he had been strongly prohibited in more than one dream4 ,for ,syrianus appearing to him in his sleep1 had deterred him with threats from the design4 ,having therefore employed other machines1 ,i intreated that at least he would mark what he principally approved of in the books of his master2 which when this best of men had performed1 in consequence of my persuasions1 and had noted some things in the front of each of ,syrianus's commentaries1 we obtained a collection of all these1 and by this means scholia1 and commentaries of no small bulk2 though to accomplish this on the whole of that divine poetry1 and on all the ,orphic rhapsiodies1 was not the intention of ,proclus4 ,but since1 as we have said1 his incredible study of these concerns1 procured him a greater and more perfect degree of theurgic virtue1 hence he no longer remained in the contemplative order1 nor contented with either of the twofold properties in divine concerns1 exercised intellect and thought alone on the more excellent objects of speculation3 but he was careful to obtain another kind of virtues more divine than the former1 and separated from the politic mode2 for he used the ,chaldean assemblies and conferences1 and their divine and ineffable concealments4 ,and having comprehended these1 he learned the manner of pronouncing and expressing them1 with their remaining use1 from ,asclepigenia1 the daughter of ,plutarch3 for she alone1 at that time1 preserved the knowledge of the great ,orgies1 and of the whole theurgic discipline1 delivered to her by her father1 who received it from ,nestor4 ,besides this1 our philosopher also being purified in an orderly manner in the ,chaldean lustrations1 was an inspector of the lucid hecatic phasmai 7or visions7 of which he himself makes mention in one of his commentaries4 ,but by moving a certain hecatic sphrula#fc1 he very opportunely brought down showers of rain1 and freed ,athens from an unseasonable heat4 ,besides this1 by certain phylacteria1 or charms1 he stopt an earthquake1 and had thoroughly tried the energies of a divining tripod1 having learned from certain verses concerning its defect4 ,for when he was in his fortieth year1 he thought in a dream1 that he repeated to himself1 the following verses3 ,high above ther there with radiance bright1 ,a pure immortal splendor wings its flight2 ,whose beams divine with vivid force aspire1 ,and leap resounding from a fount of fire4 ,and in the beginning of his forty-second year1 he appeared to himself to pronounce these verses1 with a loud voice3 ,lo6 on my soul a sacred fire descends1 ,whose vivid pow'r the intellect extends2 ,from whence far-beaming through dull bodies night1 ,it soars to ther deck'd with starry light2 ,and with soft murmurs through the azure round1 ,the lucid regions of the gods resound4 ,besides1 he clearly perceived that he belonged to the ,mercurial series2 and was persuaded from a dream that he possessed the soul of#fd ,nicomachus the ,pythagorean4 ,indeed1 if ,i were willing to be prolix1 ,i could relate many theurgical operations of this blessed man2 from the infinite number of which1 ,i will only relate one in this place of a very wonderful nature4 ,asclepigenia1 the daughter of ,archiades and ,plutarch1 but the wife of ,theagenes1 from whom ,i have received many benefits1 while she was yet a girl1 and was nourished by her parents1 fell into a disease1 grievous1 and incurable by the physicians4 ,and ,archiades1 who had reposed in her alone the hope of his race1 was deeply afflicted with her condition4 ,but the physicians despairing of her recovery1 the father fled to his last anchor1 ,proclus the philosopher1 which he was accustomed to do in affairs of the greatest moment1 considering him as his best preserver1 and earnestly intreated him to intercede by his prayers with the god for his daughter1 whose condition was well known to our philosopher4 ,proclus therefore1 taking with him ,pericles#fe the ,lydian1 a man well deserving the name of a philosopher1 went to the temple of ,esculapius#ff1 that he might pray to the god for the health of the sick virgin3 for the city at that time happily possessed this divinity1 and rejoiced in the temple of the saviour god1 which was as yet free from the destruction of the ,christians4 ,as soon1 therefore1 as ,proclus had prayed after the ancient manner1 the girl immediately perceived a great change and alleviation of her disease2 for the preserving god had easily restored her to health4 ,the sacred rites being finished1 ,proclus went to ,asclepigenia1 and found her delivered from the molestations of disease1 and in a healthy condition4 ,but this affair1 with many others1 was performed privately1 so that no traces of it remained to succeeding investigators2 and the house in which he resided greatly assisted him in this design4 ,for besides his other fortunes1 he enjoyed a most convenient dwelling1 which his father ,syrianus and his grand-father ,plutarch 7for so he did not hesitate to call him7 had once inhabited2 and this was situated near the temple of ,esculapius1 together with that of the celebrated ,sophocles1 and of ,bacchus1 which is next to the theatre1 and looks towards1 or is at least seen from1 the lofty towers of ,minerva4 ,but how much ,proclus was loved by the philosophic goddess is abundantly evinced by his philosophic life1 which he chose through her persuasions1 and that with the great success we have hitherto described4 ,but she clearly demonstrated her affection to ,proclus1 by the following circumstance4 ,when her image1 which had been so long dedicated in the parthenon1 or temple1 was taken away by those#fg who1 without any hesitation1 moved out of their places things the most holy1 and which ought to be immoveable1 there appeared to the philosopher in a dream1 a woman of a graceful form1 who admonished him to build a temple with great expedition1 for1 says she1 it pleases ,minerva1 the presiding deity of philosophy1 to dwell with you4 ,and how familiar he was with ,esculapius1 besides what we have mentioned above1 may be evinced from his perceiving the presence of the same god in his last illness4 ,for while he was between sleeping and waking1 he perceived a dragon creeping on his head1 which vision occasioned a remission of his disease1 and a mitigation of his pain2 so that it is probable he would have been restored to perfect health1 if his desire and vehement expectation of death had not prevented his recovery1 or his no longer bestowing a diligent attention on his body4 ,but he also related the following circumstances1 7worthy of being remembered7 but not without tears1 through the sympathy of his mind4 ,for1 when a young man1 he was afraid lest he should be infested with the gout1 which was the disease of his father1 and which loves to descend from parents to their children4 ,nor was he afraid1 as it seems1 without reason2 for prior to that which we have already related concerning him1 he was tormented with pains of this kind1 when another extraordinary circumstance happened to this blessed man4 ,in consequence of a certain person's prescription1 whom he had consulted1 he applied a plaster to his tormented foot1 which an unexpected bird flew away with as he was lying on his bed4 ,and this was certainly a divine and salutary symbol to ,proclus1 and might have persuaded him not to be afraid of this calamity in future4 ,but he1 as ,i have said1 was notwithstanding afflicted with the dread of this disease2 he therefore supplicated the healing god concerning this1 and intreated him to afford him a more evident token of his will3 and after this he saw in his sleep 7it is bold indeed to conceive such a circumstance in the mind1 but we must dare1 nevertheless1 nor dread to bring truth to light7 ,esculapius approaching to him1 and accurately contemplating his leg1 nor disdaining1 through his great philanthropy to embrace his knee4 ,hence1 in consequence of this vision1 ,proclus was1 through the whole of his life1 free from the apprehensions of this disease1 and was never more tortured with this kind of pains4 ,again1 the god who is worshipped by the ,adrotteni1 did not less conspicuously evince his great familiarity with this friend of the gods2 for1 more than once1 the god benevolently presented himself to the sight of ,proclus1 frequenting his temple4 ,and when he was doubtful1 and wished to know what god or gods resided and were worshipped in that place1 because the inhabitants were of different opinions in this affair1 some believing that the temple of ,esculapius was there1 persuaded of this by various signs1 as voices are certainly said to be heard in that place1 and a table is consecrated to the gods1 and oracles are given concerning the recovery of health1 and those who approach are miraculously delivered from the most grievous dangers of life4 ,others1 on the contrary1 think that the ,dioscuri reside in that place1 because there have been those who saw in the way tending towards ,adrotta1 two young men of the most beautiful form1 and riding with great celerity on horseback2 at the same time declaring1 that they hastened to the temple4 ,they add besides1 that the countenance of these was indeed human1 but that they immediately gave evidence of a more divine presence2 for when the men were arrived at the temple1 the above mentioned youths appeared to them1 making no enquiry concerning the affair1 and occupied in the sacred concerns2 but presently after1 withdrew themselves from their sight4 ,as the philosopher1 therefore1 was in doubt concerning these divinities1 and did not discredit the relations1 having requested the gods who inhabited that place1 that they would condescend to manifest who they were2 a god appeared to him in a dream1 and clearly spoke to him as follows3 ,what6 ,hast thou not heard of ,jamblichus teaching concerning1 and celebrating those two1 ,machaon and ,podalirius8 ,and besides this1 the god condescended to afford so great an instance of benevolence to this blessed man1 that he stood after the manner of those who bestow encomiums on others in the theatres1 and with a clear voice1 and composed habit1 extending his right hand1 did not hesitate to exclaim1 7that ,i may relate the words of the god17 ,proclus the ornament of the city4 ,and what could possibly be a greater testimony of this blessed man's friendship with divinity8 ,but he1 indeed1 in consequence of a most remarkable sympathy1 by which he was united with the gods1 could never refrain from tears when he mentioned this affair to us1 and related the divine encomium of himself4 ,indeed1 if ,i were willing to pursue every particular in like manner1 and to relate his familiarity with the ,hermetic ,pan1 together with the great benevolence and manifold assistance which that divinity condescended to afford ,proclus at ,athens1 and of that perfectly singular kind of felicity which he obtained from the mother of the gods1 and in which he was accustomed especially to triumph and rejoice2 ,i should perhaps seem to many readers1 to be rash in my assertions1 and to others1 the author of things perfectly incredible4 ,for many and great were the daily instances of this goddess's benevolence towards him1 in words and actions1 which are both innumerable and unheard of1 and concerning which1 ,i do not at present retain an accurate remembrance4 ,but if any one is desirous of knowing how great he was in these1 he must evolve his book concerning the mother of the gods1 from which he will understand1 that it was not without divine assistance he produced all the theology of that goddess into light2 as likewise1 whatever is acted or affirmed concerning the same in the fables of ,athis1 which he has explained after a philosophical manner1 that vulgar ears might afterwards cease to be disturbed1 on perceiving the lamentations and other obscure ceremonies with which her mysteries are celebrated4 ,after having1 therefore1 run through the theurgic virtue of ,proclus1 and its energies1 and the happy circumstances which befell him1 respecting its operations1 and having shewn1 that he did not less excel in every kind of virtue1 and that he was a man such as mortals have not beheld for a long period of time1 it remains that we now bring our discourse concerning him to a conclusion3 for we are not now at the beginning of our narration1 nor does the half of the whole remain to us1 according to the proverb1 but the whole itself is now perfect and complete4 ,since having begun from the felicity of the philosopher1 and proceeded in its exemplification1 our discourse now returns to it again4 ,for we have explained the great goods1 and providential exertions1 which were granted to this most excellent man from the gods1 and have shewn their prompt attention to his prayers1 the divine visions which he enjoyed1 and the help and solicitude which the gods testified towards him4 ,we have likewise explained his prosperous fate1 and propitious fortune1 his country1 parents1 the good habit of his body1 his masters1 friends1 and other external advantages2 each of which we have shewn to have been far greater and more splendid in him than in other men2 and have likewise diligently enumerated such things as cannot be reckoned amongst outward allurements1 but entirely depended on his will1 such as are the upright and illustrious deeds of his soul1 according to universal virtue4 ,and thus we have summarily demonstrated that his soul arrived in reality at the summit of the most consummate virtue1 and was happily established in a perfect life1 by human and divine goods of every kind4 ,but that the lovers of more elegant studies way be able to conjecture from the position of the stars under which he was born1 that the condition of his life1 was by no means among the last or middling classes1 but ranked among the highest and most happy orders1 we have thought fit to expose in this place1 the following scheme of his nativity#fh4 #af #bf' #ag #bi #bd #bc #bd #da #bi #ej -- #bc #d #db ,horos4 #h #ai ,mid4 ,heaven4 #d #db or the head of the dragon4 #bd #cc ,the new moon preceding his birth4 #h #ea ,but ,proclus departed from this corporeal life1 in the one hundred and twenty-fourth year after the government of ,julian1 on the seventeenth day of the ,attic ,munichion1 or the ,april of the ,romans4 ,nicagoras1 the junior1 being at that time the ,athenian archon4 ,the dead body was taken care of according to the ,athenian rites1 as he himself while living had directed4 ,and if that diligent attention which is due to the deceased1 was ever paid by any one1 it was certainly bestowed by this most blessed man4 ,for he did not neglect any particular of the accustomed ceremonies1 but every year1 on certain days1 visited the sepulchres of the ,athenian heroes and philosophers1 and besides this sacrificed himself1 and not through the medium of another1 to the manes of his friends and familiars4 ,and having thus exhibited to each a proper reverence and honour1 proceeding into the ,academy1 he there1 in a certain place1 pleased one by one the manes of his ancestors and kindred2 and shortly after1 in another part of the same ,academy1 he supplicated in common the souls of all philosophers4 ,and these ceremonies being ended1 this most excellent man1 having chosen a third place1 performed sacred rites to all the souls of deceased mankind4 ,the dead body of ,proclus1 therefore1 being disposed of according to his appointment1 as we have said1 was carried by his friends and buried in the eastern part of the suburbs#fi1 near to ,lycabetus1 where also the body of his preceptor ,syrianus reposes1 who1 when he was alive1 had requested this of our philosopher1 and1 in consequence of this1 had taken care to procure a twofold receptacle in his sepulchre4 ,but when this most holy man was consulted how he would wish his funeral to be conducted1 lest there should be any thing indecent1 or without a proper decorum1 he desired flutes1 with which he was threatened in a dream1 and nothing besides4 ,an epigram1 consisting of four verses1 is inscribed on his tomb1 which he composed himself1 as follows3 ,i ,proclus1 here the debt of nature paid1 7,my country ,lycia7 in the dust am laid2 ,great ,syrianus form'd my early youth1 ,and left me his successor in the truth4 ,one common tomb1 our earthly part contains1 ,one place our kindred souls1--th' therial plains4 ,nor were prognostics wanting in the year prior to his decease1 such as an eclipse#gj of the sun1 so great that night was produced in the day-time3 for the darkness was so profound1 that the stars became visible4 ,and this happened when the sun was in ,capricorn1 in the eastern centre4 ,besides1 the writers of ,ephemeride observe that there will be another eclipse of the sun1 at the conclusion of the next year4 ,but such like affections of the heavenly bodies are said to portend events on the earth3 to us indeed1 the eclipse perspicuously signified a privation and1 as it were1 defection of the illustrious luminary of philosophy4 ,and thus much may suffice for the life of the philosopher4 ,it now remains for those who are willing to undertake the labour1 to write concerning his disciples and friends4 ,for it appears that many resorted to him from all parts2 among which some were only his auditors1 but others firm adherents to his doctrine2 and1 on account of philosophy1 his familiars4 ,i could likewise wish that some one1 who is more laborious than myself1 would give a particular account of his writings4 ,for ,i have alone delivered these particulars of his life1 that ,i might satisfy my conscience1 and might testify that ,i religiously reverence the soul of ,proclus1 and his good dmon4 ,i shall1 therefore1 add nothing concerning the philosopher's writings1 except that ,i have always heard him prefer his commentaries on the ,timus of ,plato beyond all his works4 ,he likewise vehemently approved of his commentaries on the ,thetetus4 ,and he used frequently to say1 that if he was endued with despotic power1 he would alone preserve1 of all the writings of the ancients1 the ,oracles and the ,timus4 ,for1 said he1 ,i would abolish all the rest1 and remove them from the present age1 because it happens that many are offended1 who undertake to read them rashly1 and without proper institution4 ,concerning the ,published ,writings of ,,proclus4 ,marinus having neglected to give us an account of the writings of ,proclus1 ,i thought it would not be unacceptable to the reader1 to present him with a catalogue of his works which are still extant2 and the most perfect relation ,i am able of such as are unfortunately lost4 ,in the execution of this design1 ,i shall follow1 for the most part1 the accurate ,fabricius in the order1 and critical account of his works2 but shall not hesitate to dissent from him in deciding on their philosophical merit3 for it is very rare that philology and philosophy are united in the same person1 and coalesce in amicable conjunction4 99 99 99 99 99 #a4 ,four ,hymns4 ,in the preceding life we are informed1 by ,marinus1 that ,proclus composed many hymns1 in which the divinities1 both of ,greeks and ,barbarians1 were celebrated2 but unfortunately there are only four preserved1 the first of which is to the ,sun1 the second and third to ,venus 7which may be found in the ,florilegium of ,stobus1 p4 #bdi4 edit4 ,grotii17 and the fourth to the ,muses4 ,they are collected by ,fabricius1 in the eighth volume of his ,bibliotheca ,grca2 and are most happy specimens of philosophical poetry4 ,indeed1 they bear most evident marks of a mind full of divine light1 and agitated with the fury of the ,muses2 and possess all that elegance of composition for which the writings of ,proclus are so remarkable4 ,so that it is very strange ,gyraldus should ascribe them to a ,hierophant of ,laodica1 of that name2 since1 as ,fabricius observes1 ,suidas mentions other writings of this priest1 but does not speak of him as the author of any hymns4 ,and if he had1 these hymns breathe too much of the spirit and manner of ,proclus1 to be the production of any other4 #b4 ,two ,books concerning the useful ,parts of ,learning1 contained in ,grammar4 ,three books on this subject are mentioned by ,suidas1 and four by ,photius2 but there are alone extant fragments of two1 in the ,bibliotheca of ,photius2 which were published separate by ,andrew ,schottus1 at the end of the syntax of ,apollonius ,alexandrinus1 ,francof4 #aeij4 ,hanov4 #afae4 quarto4 ,it appears from these extracts1 that ,proclus1 in this work1 not only treats of every kind of verse1 but also of the more celebrated poets4 ,and hence ,fabricius conjectures1 that the short life of ,homer1 which ,leo ,allatius published under the name of ,proclus1 was taken from the first book of this ,chrestomathia4 #c4 ,eighteen ,arguments against the ,christians4 ,the learned ,cave1 was egregiously mistaken1 in supposing that this work1 as well as the answer to it by ,philoponus1 is lost2 not considering1 probably1 that these arguments in defence of the world's eternity are 7as ,fabricius well observes7 directly against one of the ,christian dogmata1 the creation of the world4 ,these arguments 7except the first1 which is lost17 are preserved in the answer of ,philoponus2 the ,greek edition of which was published at ,venice1 in #aece1 folio2 and in ,latin from the version of ,joh4 ,mahatius1 at ,lyons1 #aeeg4 folio4 ,simplicius1 in his commentaries on the first book of ,aristotle de ,clo1 and elsewhere1 opposes this work of ,philoponus2 at the same time1 suppressing his name4 ,the arguments are1 in my opinion1 extremely subtle1 and unanswerable2 and ,philoponus1 in his refutation1 every where discovers1 that he has by no means fathomed the depth of our philosopher4 #d4 ,a ,commentary on the ,timus of ,plato1 in five ,books4 ,a most admirable work1 composed in the flower of his age1 when he was twenty-eight years old1 according to ,marinus4 ,this invaluable treasure1 may be said to contain all the wisdom of the highest antiquity3 for ,proclus every where cites the most celebrated interpreters on the ,timus1 such as ,adrastus the ,peripatetic2 lianus and ,albinus1 ,platonists2 ,aristocles1 ,clearchus1 ,crantor1 ,dercyllides2 and especially ,jamblichus2 likewise ,longinus1 whom he often refutes2 ,numenius the ,pythagorean1 and ,origen1 7different from ,adamantius17 together with ,plotinus1 ,porphyry the ,platonic1 ,ptolemy1 ,severus1 and ,theodorus ,asinus2 but he always prefers the explications of his master ,syrianus1 before all the rest4 ,add too1 that he every where conciliates the ,orphic writings and the ,chaldean oracles with the theology of ,plato4 ,and with respect to ,orpheus in particular1 we are indebted to these sublime commentaries for the greatest part of the fragments of that divine poet4 ,he is perpetually on the wing1 and rises gradually in elevated conceptions1 until his mind1 like ,homer's discord1 reaches the heavens4 ,his style is at the same time nervous and diffuse1 accurate and elegant4 ,we are astonished with the magnificence of his metaphors1 delighted with the copious variety of his matter1 and filled with a divine light1 by the sacred truths he unfolds4 ,this great work1 however1 unfortunately1 scarcely explains a third part of the ,timus2 from whence it is probable1 as ,fabricius well observes1 that several books have been lost through the injury of time4 ,it was published in ,greek1 at ,basil1 in the year #aeef2 and is full of mistakes1 as is generally the case with the ,basil editions of books1 so that a more correct edition is greatly to be desired1 though at present not much to be expected4 ,on concluding my account of this inestimable work1 ,i find my indignation roused by the following words of ,dr4 ,cudworth1 in his ,intellectual ,system1 p4 #cjf4 8,proclus 7says he7 had some peculiar fancies and whims of his own1 and was indeed a confounder of the ,platonic theology1 and a mingler of much unintelligible stuff with it40 ,i must confess1 7and ,i am neither afraid1 nor ashamed of the declaration17 that ,i never found any thing in ,proclus1 but what by patient thought1 accompanied with a sincere and vehement thirst after truth1 ,i have been able to fathom4 ,had ,dr4 ,cudworth been endued with these requisites1 he would doubtless have had equal success2 but without them1 the sublimest truths will certainly appear to be unintelligible stuff4 ,besides this consideration is not to be omitted1 that a modern priest makes a bad philosopher4 #e4 ,on ,plato's ,republic4 ,these commentaries1 or rather fragments of commentaries1 are extant in ,greek1 at the end of ,proclus on the ,timus4 ,suidas mentions four books of ,proclus on ,plato's politics2 and some of ,proclus's dissertations on these books1 were found 7according to ,fabricius7 in the library of ,lucas ,holstenius4 ,the chief design of this work seems to be the unveiling the theological mysteries concealed under the fables of ,homer1 and other divine poets2 which ,proclus has accomplished 7in my opinion7 in a most wonderful manner4 ,that ,homer1 indeed1 every where abounds with ,egyptian learning1 is obvious to every one2 but few are acquainted with the profound wisdom which his fables conceal4 ,the latent meaning of most of these is unfolded in the present invaluable1 though imperfect work2 and he whose mind is sufficiently enlightened by the ancient philosophy1 to comprehend the beauty of these illustrations1 will receive an additional delight from the study of ,homer1 which it is impossible to express4 ,an epitome of this work was published in ,latin1 by the learned ,gesner1 #aedb4 #hvo4 under the following title3 ,apologi qudam pro ,homero1 et ,arte ,poetica1 ,fabularumque aliquot ,enarrationes ex commentariis ,procli ,lycii ,diadochi philosophi ,platonici in libros ,platonis de ,rep4 in quibus plurim de ,diis ,fabul non juxta grammaticorum vulgus historic1 physic aut ethic tractantur1 sed ,theologicis1 ut ,gentiles loquntur1 ex prima ,philosophia rationibus explanantur4 #f4 ,on the first ,alcibiades of ,plato4 ,ficinus translated parts of this work into ,latin1 and published them under the title of ,procli de ,anima ac dmone1 de ,sacrificio et ,magia1 ,venice #adig4 and #aeaf4 fol4 by ,aldus2 and in a more simple form at ,lyons4 ,fabricius informs us1 that the manuscript commentary of ,proclus in ,greek1 but scarcely explaining the half part of the ,alcibiades1 is to be found in various libraries of ,france1 ,england1 and ,italy4 ,also at ,lyons1 among the books of ,isaac ,vossius2 and at ,hamburgh in the ,johannean library4 ,from the specimen given of this work by ,ficinus1 it appears1 like all ,proclus's philosophical writings1 to be an invaluable treasury of wisdom2 and nothing certainly1 reflects greater disgrace on a nation than suffering such monuments of ancient learning and wisdom to lie concealed in colleges1 covered with dust1 and never consulted4 #g4 ,six ,books on ,plato's ,theology4 ,a most divine work1 in which the philosopher collects into a system the theology dispersed in the writings of ,plato1 and establishes it by invincible demonstrations4 ,he deduces1 in a beautiful and connected series1 all the divine orders1 from the retreats of ineffable unity2 every where connects them by proper mediums1 and1 after leading us through the long gradation of principles1 brings us back again to the original from whence they flowed1 and to which they constantly tend4 ,the whole is uncommonly profound and abstruse2 and it was not before the third reading1 that ,i could fathom the depth it contains4 ,fabricius observes1 8that it is a subtle and learned work1 but from which1 you will sooner learn the opinion of ,syrianus and ,proclus1 concerning the deity and divine concerns1 than that of ,plato4 ,he adds1 that it is usual with the ,platonists1 even from ,plotinus1 to unite to the doctrine of ,plato1 a thousand dogmata1 foreign from his philosophy1 as if ,plato1 though he did not perceive after this manner1 ought certainly so to perceive40 ,when men mistake their abilities1 they always act absurdly1 and often dangerously4 ,as a laborious and accurate critic on philological matters1 ,fabricius merits the highest commendation such attainments can deserve2 but when he leaves the beaten road in which nature designed him to walk1 and attempts the tractless paths of philosophy1 he perpetually stumbles1 and often falls on the ground4 ,the wings of philology1 like those of the swallow1 were never destined for a lofty flight3--it must be the eagle wing of genius1 which can alone soar to the sun of philosophy4 ,the ,greek and ,latin edition of this valuable work1 was published at ,hamburgh1 by milius ,portus1 #afah1 folio4 #h4 ,theological ,institutions2 or1 as it may be called1 the ,elements of ,theology4 ,this admirable work contains two hundred and ten propositions1 disposed in a scientific order1 and supported by the firmest demonstrations4 ,they begin from super-essential unity1 and proceed gradually through all the beautiful and wonderful progressions of divine causes1 ending in the self-moving energies of soul4 ,they possess all the accuracy of ,euclid1 and all the subtilty and sublimity necessary to a knowledge of the most profound theology2 and may be considered as bearing the same relation to the ,pythagoric and ,platonic wisdom1 as ,euclid's ,elements1 to the most abstruse geometry4 ,patricius1 the first ,latin translator of this divine work1 seems to have been very sensible of the truth of this observation3 for he every where carefully distinguishes the propositions from their demonstrations2 and adds the word corollary to such consequencies as merit that appellation4 ,his edition was published at ,ferraria1 #aehc4 quarto1 under the title of ,theological ,elements4 ,the ,greek and ,latin edition1 is subjoined to ,proclus's six books on ,plato's ,theology1 ,hamburgh #afah4 folio4 #i4 ,two ,books concerning ,motion4 ,this useful work1 collected1 as ,fabricius observes1 from the third and following books of ,aristotle's physics1 was published in ,greek at ,basil1 #aeca1 and with the ,latin version of one ,justus ,velsius1 a physician1 ,basil1 #aede4 octavo4 ,it was likewise translated by ,patricius1 and is annexed to his version of the ,theological ,institutions4 #aj4 ,an ,hypotyposis1 or ,information concerning ,astronomical ,hypotheses4 ,this work1 which ,fabricius observes is a compendium of ,ptolemy's ,almagest1 was published in ,greek1 at ,basil1 #aedj4 quarto2 and in ,latin by ,george ,valla1 folio1 #aeda4 ,a part of this work1 which treats of the use of the astrolabe1 ,fabricius informs us1 is extant in manuscript1 in various libraries4 ,the same accurate critic likewise observes1 that a small treatise1 inscribed ,uranodromus1 is extant1 under the name of ,proclus1 in some libraries1 as in that of ,vindobona1 and of ,oxford1 among the ,barrocian volumes4 ,the comprehensive variety of ,proclus's genius equally demands our admiration and applause4 #aa4 ,a small ,treatise concerning the ,sphere1 or ,celestial ,circles4 ,this little work is an accurate and elegant introduction to astronomy2 and is almost wholly taken from the ,isagoge of ,geminus ,rhodius1 on the phnomena4 ,the best editions are the ,greek and ,latin one published at ,paris in #aeec1 quarto2 and that of ,bainbridge1 professor of astronomy at ,oxford1 ,london #afbj4 quarto4 #ab4 ,a ,paraphrase in four ,books1 on the ,quadripartite of ,ptolemy4 ,this elegant work must1 ,i should imagine1 be an invaluable treasure to the lovers of astrology4 ,it was first published in ,greek by ,melancthon2 and afterwards in ,greek and ,latin by ,leo ,allatius1 at ,lyons ,bat4 #afed4 octavo4 #ac4 ,four ,books1 on the first ,book of ,euclid's ,elements4 ,for an account of this work1 see the introduction1 and the following sheets1 in which it speaks for itself1 in an ,english dress4 #ad4 ,a ,commentary on ,hesiod's ,works and ,days4 ,this work contains a valuable moral explanation of this great poet's meaning2 and ,fabricius justly observes1 that he is often assaulted without occasion1 by the petulant jeers of that vain man ,joh4 ,tzetzes4 ,the best edition of this work is that of ,daniel ,heinsius1 ,lugd4 ,bat4 #afjc4 quarto4 #ae4 ,fabricius informs us1 that in some manuscripts1 as in the ,vindobonensian and ,barrocian1 a small treatise is usually ascribed to ,proclus1 entitled ,epistolic ,characters2 and is prefixed to the ,epistles of ,phalaris and ,brutis1 and published under the name of ,libanius1 in ,greek1 with the version of ,casp4 ,stibilinus1 ,commelin4 #aeig1 octavo4 ,but it is doubtful whether ,proclus is the genuine author of this work3 from the title1 ,i should suppose the contrary4 ,and thus much for an account of those writings of ,proclus which have escaped the ravages of time1 and have been fortunately exposed to public inspection3 it now remains that we relate such inestimable works of this philosopher1 as are yet preserved in shameful concealment2 or are utterly lost in the ruins of antiquity4 ,concerning the ,unpublished ,writings of ,,proclus4 #af4 ,on the ,alcibiades of ,plato4 ,see num4 #f4 #ag4 ,on ,plato's ,politics4 ,see num4 #e4 #ah4 ,on ,plato's ,parmenides4 ,a commentary1 in seven books2 the last of which was not completed by ,proclus1 but by ,damascius4 ,from occasional fragments1 which have been published of this commentary1 it appears to be a most divine work2 and indeed it cannot be otherwise1 if we consider it as the production of one of the greatest philosophers1 on the most sublime and profound of all ,plato's ,dialogues4 ,it is dedicated to ,asclepiodotus1 a physician and philosopher1 and is not only extant in ,greek ,,ms4 in the library of the ,german emperor1 according to ,lambecius1 lib4 vii4 p4 #da4 but also in ,latin1 from the unpublished version of one ,antonius ,hermannus ,gogava1 as the same ,lambecius informs us1 p4 #da4 ,four books of this work are extant in ,greek1 in the ,bodleian library at ,oxford2 and it is much to be lamented that ,thomson did not publish these1 instead of his trifling edition of the ,parmenides4 ,fabricius likewise informs us1 that ,livius ,galantes mentions his having found six of these books in some of the ,italian libraries4 ,they are also extant in the ,medican library of the great ,etruscan commander4 #ai4 ,on the ,cratylus of ,plato4 ,we have already observed1 in the dissertation on the ,orphic theology1 p4 #aje4 what a great treasure of ancient mythology1 must be contained in this work2 but there is little hope of its ever emerging from the obscurity of public libraries4 ,it is extant in ,greek1 not only in the ,italian libraries1 but also among the manuscript books of ,isaac ,vossius4 #bj4 ,uranodromus4 ,see above1 num4 #aj4 #ba4 ,ten ,doubts concerning ,providence1 in one ,book4 ,philoponus mentions this work1 in his second book against ,proclus on the eternity of the world2 and a ,latin version of it is extant by one ,william de ,morbeka1 in the ,johannean library of ,hamburgh4 ,extracts from this translation are preserved by ,fabricius1 in his ,greek ,library2 and they are in every respect worthy of the genius of ,proclus4 #bb4 ,concerning ,providence and ,fate1 and that which is in our ,power1 one ,book4 ,this work is dedicated to one ,theodorus1 a mechanist2 and is extant in the ,latin translation of the same ,morbeka1 in the ,greek ,library of ,fabricius4 ,the translation is for the most part barbarous1 but is1 however1 sufficiently legible to discover that it is a most valuable treatise1 replete with the usual elegance1 subtilty1 and sublimity of our philosopher4 #bc4 ,concerning the ,hypostasis1 or ,subsistence of ,evil4 ,this book is extant in ,latin1 in the ,johannean library2 and fragments of it are preserved by ,fabricius1 in his ,greek library4 ,it is to be regretted1 that ,fabricius did not preserve the whole in that excellent philological work4 ,concerning the ,lost ,writings of ,,proclus4 #bd4 ,on the ,speech of ,diotima1 in ,plato's ,banquet1 concerning the ,subsistence of the ,beautiful4 ,fabricius informs us1 that this work is distributed into many books2 and ,holstenius observes1 that it is mentioned in a certain scholium of the ,medicean copy of ,proclus's commentaries on ,plato's politics2 but it is unfortunately no where extant4 #be4 ,on the ,philebus of ,plato2 as may be inferred from the narration of ,damascius in ,photius1 p4 #eej2 and ,suidas in ,marinus4 ,for ,damascius relates1 that ,marinus having composed a commentary on this dialogue1 on shewing it to ,isidorus for his approbation1 that philosopher observed1 that those of his master were sufficient2 which words ,fabricius1 with great propriety1 applies to the commentaries of ,proclus on the ,philebus4 #bf4 ,on the ,thetetus of ,plato4 ,this work is praised by ,marinus1 in the last chapter of the preceding life2 and no doubt with great propriety3 for this abstruse and sublime dialogue would naturally call forth all the divine fire and elegance of our philosopher4 #bg4 ,commentaries on the ,enneads of ,plotinus4 ,this work is mentioned by ,gyraldus1 in his second dialogue on ancient poets2 by ,ficinus on ,plotinus2 by ,philip ,labbeus1 in his account of ,,ms4 books1 p4 #bhf2 and in the notes of ,bullialdus to ,theo of ,smyrna1 p4 #bbd4 ,but also in a certain note prefixed to an ancient manuscript of ,jamblichus1 on the ,egyptian mysteries1 to this effect3 8,the philosopher ,proclus1 commenting on the ,enneads of the great ,plotinus1 says1 that it is the divine ,jamblichus who answers the epistle of ,porphyry40 ,this note is in ,greek1 in the original1 and is 7in my opinion7 of itself sufficient to prove that such a work was once extant1 though now unfortunately lost4 ,how much the want of these commentaries is to be regretted1 must be deeply felt by every lover of the ,platonic philosophy4 ,for the unequalled profundity1 and divine mysteries1 contained in the writings of ,plotinus1 could never be more happily illustrated than by the irradiations of such a genius as ,proclus4 #bh4 ,lectures on ,aristotle's ,book 1 or concerning ,interpretation4 ,this work1 it seems1 was never published2 but ,ammonius ,hermeas1 the disciple of ,proclus1 has inserted in his valuable commentary on this book all that he could retain in his memory of ,proclus's lectures4 #bi4 ,hymns1 not a few1 see num4 #a4 #cj4 ,chrestomathea4 ,see num4 #c4 #ca4 ,on the ,mother of the ,gods1 one book1 mentioned by ,marinus1 in the preceding ,life4 #cb4 ,on the ,theology of ,orpheus4 ,this work is mentioned by ,marinus1 in the preceding ,life1 and by ,suidas2 and its loss must be particularly regretted by all the lovers of recondite theology4 #cc4 ,ten ,books1 on the ,chaldean ,oracles4 ,this most valuable work is mentioned by ,marinus1 in the preceding ,life1 and by ,proclus himself on ,plato's ,politics1 p4 #cei4 ,it was doubtless not extant at the time when ,psellus and ,pletho undertook the illustration of a few of these oracles3 at least the inconsiderable merit of their commentaries1 strongly favours this supposition4 #cd4 ,a ,commentary on the whole of ,homer4 ,suidas4 ,a specimen of the great value of this work may be seen in our philosopher's commentaries on ,plato's republic4 ,the works of ,homer are not only the great fountain of poetry1 but likewise of philosophy2 and are no less admirable for inspiring the fury of the ,muses than for containing the mysteries of the most recondite theology4 #ce4 ,concerning the ,gods1 according to ,homer4 ,had this work been preserved1 we should doubtless have been furnished with a defence of the heathen religion1 which would have silenced the ignorant clamours of its opponents4 #cf4 ,the ,symphony or ,concord of ,orpheus1 ,pythagoras1 and ,plato4 ,suidas4 ,proclus1 in his published writings1 is every where studious of reconciling the doctrines of these great men1 and is always successful in this undertaking4 ,indeed1 the same divine genius seems to have irradiated and inspired these wonderful heroes1 but in different ways3 in ,orpheus it was accompanied with the fire of the ,muses2 in ,pythagoras it shone through the mysterious veil of numbers2 and in ,plato1 combining the preceding modes1 it was seen enshrined in awful majesty of thought1 clothed with the graces of poetical diction1 and resplendent with ineffable light4 #cg4 ,two ,books on the ,theurgic ,discipline4 ,suidas4 ,how much ,proclus excelled in this art1 may be seen in the preceding ,life4 #ch4 ,concerning the ,oppositions of ,aristotle to ,plato's ,timus4 ,this work is mentioned by ,proclus in the #c;d book of his commentary on the ,timus1 p4 #bbf4 and seems to have escaped the notice of the accurate ,fabricius4 ,aristotle may1 no doubt1 in many particulars be reconciled with ,plato2 but it is also certain1 that in some he is perfectly dissonant4 ,and thus much for the ,life and ,writings of ,proclus4 ,,commentaries ,,of ,,proclus4 ,,book ,i4 ,,chap4 ,i4 ,on the ,middle ,nature of the ,mathematical ,essence4 ,it is necessary that the mathematical essence should neither be separated from the first nor last genera of things1 nor from that which obtains a simplicity of essence2 but that it should obtain a middle situation between substances destitute of parts1 simple1 incomposite and indivisible1 and such as are subject to partition1 and are terminated in manifold compositions and various divisions4 ,for since that which subsists in its inherent reasons remains perpetually the same1 is firm and durable1 and cannot be confuted1 it evidently declares it is superior to the forms existing in matter4 ,but that power of progression which apprehends1 and which besides uses the dimensions of subjects1 and prepares different conclusions from different principles1 gives it an order inferior to that nature which is allotted an indivisible essence1 perfectly constituted in itself4 ,hence 7as it appears to me7#ga ,plato also divides the knowledge of things which are1 into first1 middle1 and last substances4 ,and to indivisible natures1 indeed1 he attributes an intelligence1 which1 in a collective manner1 and by a certain simple power1 divides the objects of intellectual perception2 so that being divested of matter1 and endued with the greatest purity1 it apprehends things themselves1 by a certain unifying perception1 and excels the other kinds of knowledge4 ,but to divisible essences1 and such as are allotted the lowest nature1 and to all sensible beings1 he attributes opinion1 which obtains an obscure and imperfect truth4 ,but to middle essences 7and such are mathematical forms71 and to things inferior to an indivisible and superior to a divisible nature1 he attributes cogitation4 ,for this1 indeed1 is inferior to intellect1 and the supreme science dialectic2 but is more perfect than opinion1 and more certain and pure4 ,for it advances by a discursive procession1 expands the indivisibility of intellect1 and unfolds that which was involved in the unity of intellectual apprehension3 but it collects things which are divided1 and brings them back to mind4 ,hence1 as knowledges differ among themselves1 so the objects of knowledge are distinguished by nature4 ,so that intelligible essences having an uniform subsistence1 evidently excel all others4 ,but sensibles are entirely excelled by primary essences3 and mathematical natures1 and whatever falls under cogitation1 are allotted a middle order3 for they are excelled by the division of intelligibles2 but because destitute of matter1 they are superior to sensible natures2 and by a certain simple power1 they are excelled by the first2 but by a certain reason are more exalted than the last4 ,hence they possess notions of an intellectual essence1 which are more manifest than sensibles1 but which are1 at the same time1 only the images of an intellectual nature2 and they imitate divisibly the indivisible1 and1 in a multiform manner1 the uniform exemplars of things4 ,and1 that ,i may sum up the whole in a few words1 they are placed in the vestibules or entrances of primary forms1 and disclose their indivisible and prolific subsistence collected into one1 but they do not yet excel the division and composition of reasons1 and an essence accommodated to the obscurity of images2 nor are they capable of passing beyond the various notions of the soul1 endued with a discursive power1 and of adhering to intellections perfectly simple1 and purified from all material imperfection4 ,after this manner then1 is the middle nature of mathematical genera and forms to be understood2 as filling up the medium between essences entirely indivisible1 and such as are divisible about matter4 ,,chap4 ,,ii4 ,concerning the common ,principles of ,beings1 and of the ,mathematical ,essence1#gb bound and infinite4 ,but it is necessary that1 considering the principles of the whole mathematical essence1 we should return to those general principles1 which pervade through and produce all things from themselves1 ,i mean bound and infinite4 ,for from these two after that cause of one1 which can neither be explained1 nor entirely comprehended1 every other thing1 as well as the nature of the mathematical disciplines1 is constituted4 ,in the former1 indeed1 producing all things collectively and separately2 but in these proceeding in a convenient measure1 and receiving a progression in a becoming order2 and in some1 subsisting among primary1 but in others among middle1 and in others again among posterior natures4 ,for intelligible genera1 by their simplicity of power1 are the first participants of bound and infinite3 because1 on account of their union and identity1 and their firm and stable existence1 they are perfected by bound3 but on account of their division into multitude1 their copious power of generation1 and their divine diversity and progression1 they obtain the nature of infinite4 ,but mathematical genera originate1 indeed1 from bound and infinite1 yet not from primary1 intelligible1 and occult principles only2 but also from those principles which proceed from the first to a secondary order1 and which are sufficient to produce the middle ornaments of beings1 and the variety which is alternately found in their natures4 ,hence1 in these also1 the reasons and proportions advance to infinity1 but are restrained and confined by that which is the cause of bound4 ,for number rising from the retreats of unity1 receives an incessant increase1 but that which is received as it stops in its progression1 is always finite4 ,magnitude also suffers an infinite division1 yet all the parts which are divided are bounded1 and the particles of the whole exist finite in energy4 ,so that without the being of infinity1 all magnitudes would be commensurable1 and no one would be found but what might either be explained by words1 or comprehended by reason 7in which indeed geometrical subjects appear to differ from such as are arithmetical27 and numbers would be very little able to evince the prolific power of unity1 and all the multiplex and super-particular proportions which they contain4 ,for every number changes its proportion1 looking back upon1 and diligently enquiring after unity1 and a reason prior to itself4 ,but bound being taken away1 the commensurability and communication of reasons1 and one and the same perpetual essence of forms1 together with equality1 and whatever regards a better co-ordination1 would never appear in mathematical anticipations3 nor would there be any science of these2 nor any firm and certain comprehensions4 ,hence then1 as all other genera of beings require these two principles1 so likewise the mathematical essences4 ,but such things as are last in the order of beings1 which subsist in matter1 and are formed by the plastic hand of nature1 are manifestly seen to enjoy these two principles essentially4 ,infinite as the subject seat of their forms2 but bound as that which invests them with reasons1 figures1 and forms4 ,and hence it is manifest that mathematical essences have the same pre-existent principles with all the other genera of beings4 ,,chap4 ,,iii4 ,what the common ,theorems are of the ,mathematical ,essences4 ,but as we have contemplated the common principles of things1 which are diffused through all the mathematical genera1 after the same manner we must consider those common and simple theorems1 originating from one science1 which contains all mathematical knowledge in one4 ,and we must investigate how they are capable of according with all numbers1 magnitudes and motions4 ,but of this kind are all considerations respecting proportions1 compositions1 divisions1 conversions1 and alternate changes3 also the speculation of every kind of reasons1 multiplex1 super-particular1 super-partient1 and the opposite to these3 together with the common and universal considerations respecting equal and unequal1 not as conversant in figures1 or numbers1 or motions1 but so far as each of these possesses a common nature essentially1 and affords a more simple knowledge of itself4 ,but beauty and order are also common to all the mathematical disciplines1 together with a passage from things more known1 to such as are sought for1 and a transition from these to those which are called resolutions and compositions4 ,besides1 a similitude and dissimilitude of reasons are by no means absent from the mathematical genera3 for we call some figures similar1 and others dissimilar2 and the same with respect to numbers4 ,and again1 all the considerations which regard powers1 agree in like manner to all the mathematical disciplines1 as well the powers themselves1 as things subject to their dominion3 which1 indeed1 ,socrates1 in the ,republic1 dedicates to the ,muses1 speaking things arduous and sublime1 because he had embraced things common to all mathematical reasons1 in terminated limits1 and had determined them in given numbers1 in which the measures both of abundance and sterility appear4 ,,chap4 ,,iv4 ,how these ,common ,properties subsist1 and by what ,science they are considered4 ,but it is requisite to believe1 that these common properties do not primarily subsist in many and divided forms1 nor originate from things many and the last3 but we ought to place them as things preceding in a certain simplicity and excellence4 ,for the knowledge of these antecedes many knowledges1 and supplies them with principles2 and the multitude of sciences subsist about this1 and are referred to it as their source4 ,thus the geometrician affirms1 that when four magnitudes are proportional1 they shall be alternately proportional2 and he demonstrates this from principles peculiar to his science1 and which the arithmetician never uses4 ,in like manner1 the arithmetician affirms1 that when four numbers are proportional1 they shall be so alternately3 and this he evinces from the proper principles of his science4 ,for who is he that knows alternate ratio considered by itself1 whether it subsists in magnitudes or in numbers8 ,and the division of composite magnitudes or numbers1 and in like manner1 the composition of such as are divided8 ,for surely it cannot be said that there are sciences and cognitions of things divisible3 but that we have no science of things destitute of matter1 and which are assigned a more intellectual contemplation2 for the knowledge of these is by a much greater priority science1 and from these the common reasons of many sciences are derived4 ,and there is a gradual ascent in cognitions from things more particular to more universal1 till we revert to the science of that which is1 considered as it is1 abstracted from all secondary properties4 ,for this sublime science does not think it suitable to its dignity1 to contemplate the common properties which are essentially inherent in numbers1 and are common to all quantities2 but it contemplates the one1 and firm essence of all the things which are4 ,hence1 it is the most capacious of all sciences1 and from this all the rest assume their own peculiar principles4 ,for the superior sciences always afford the first suppositions of demonstrations to such as are subordinate4 ,but that which is the most perfect of all the sciences1 distributes from itself principles to all the rest1 to some indeed1 such as are more universal1 but to others1 such as are more particular4 ,hence1 ,socrates1 in the ,thetetus1 mingling the jocose with the serious1 compares the sciences which reside in us to doves3 but he says they fly away1 some in flocks1 but others separate from one another4 ,for such1 indeed1 as are more common and more capacious1 comprehend in themselves many such as are more particular3 but such as being distributed into forms1 touch things subject to knowledge1 are distant from one another1 and can by no means be copulated together1 since they are excited by different primary principles4 ,one science1 therefore1 precedes all sciences and disciplines1 since it knows the common properties which pervade through all the genera of beings1 and supplies principles to all the mathematical sciences4 ,and thus far our doctrine concerning dialectic#gc is terminated4 ,,chap4 ,v4 ,what the ,instrument is1 which judges of the ,mathematical ,genera and ,species4 ,let us now consider what that instrument is#gd1 adapted to the judgment of mathematical concerns2 and let us appoint ,plato as our guide in this affair1 who1 in his ,republic1 divides cognitions separately from such things as are the objects of knowledge2 and distributes cognitions in conjunction with things subject to knowledge4 ,for of the things which are1 some he ranks among intelligibles1 and others among sensibles4 ,and of intelligibles1 some are again pure intelligibles1 and others subject to cogitation4 ,and of sensibles1 some are purely sensibles1 but others conjectural4 ,to intelligibles1 indeed1 which are the first of the four genera1 he assigns an intelligible knowledge2 but to those which are subject to cogitation1 he attributes thought3 to sensibles1 faith2 but to conjecturals1 a conjectural or assimilatory power4 ,and he shews1 that the assimilatory power has the same proportion to sense as thought to intelligence4 ,for the conjectural power knows the spectres of sensible forms1 while they are beheld in water and other bodies1 which perspicuously represent their image3 since1 by their situation in water1 they are after a manner1 allotted the last seat in the gradations of forms1 and truly become the resemblances of resemblances4 ,in like manner1 thought beholds the images of intelligibles in a degraded state1 fallen from primary simple and indivisible forms1 into multitude and division4 ,hence1 a knowledge of this kind1 depends on other more ancient hypotheses2 but intelligence arrives at that principle which is no longer supposed4 ,if then1 mathematical concerns are neither allotted an essence separate from all division and variety1 nor that nature which is apprehended by sense1 which is obnoxious to many mutations1 and is in every proportion divisible1 it must be manifest to every one1 that they are essentially subject to cogitation3 but cogitation presides over these as an instrument adapted to judgment1 in the same manner as sense to sensibles1 and the assimilatory power to conjecturals4 ,from whence1 indeed1 ,socrates determines that the knowledge of these is more obscure than the first science1 but is more evident than the impulsive apprehension of opinion4 ,for in this the mathematical sciences are inferior to intelligence1 because they contemplate that which is evolved1 and is endued with a power of progression2 but they are superior to opinion1 by that stability of reasons which they contain1 and which cannot be confuted4 ,and they originate from supposition1 through a diminution of the first science2 but they contain forms independent of matter1 from their possessing a knowledge more perfect than that of sensibles4 ,we have therefore determined an instrument adapted to the judgment of all mathematical concerns1 i4 e4 cogitation1 according to the mind of ,plato2 which places itself indeed above opinion1 but is excelled by intelligence4 ,,chap4 ,,vi4 ,concerning the ,essence of ,mathematical ,genera and ,species#ge4 ,it now remains1 that we consider what subsistence or essence ought to be assigned to mathematical genera and species8 ,whether we must deduce their origin and subsistence from sensible objects1 or from abstraction1 or from a collection of such things as are dispersed by parts into one common definition2 or must allow them an existence prior to that of sensibles1 as ,plato affirms1 and as the progression of universal being demonstrates8 ,first then1 if we affirm that mathematical species are composed from sensibles2 whilst the soul from material triangles or circles1 forms in herself the trigonic1 or circular species1 by a kind of secondary generation2 ,i would ask from whence is derived the great certainty and accuracy of definitions8 ,for it must either proceed from sensibles1 or from the soul herself4 ,but from sensibles is impossible1 for these1 in a continual flow of generation and decay1 do not for a moment retain an exact sameness of being2 and consequently fall far short of the exactness contained in the definitions themselves4 ,it must therefore proceed from the soul1 which1 by her immaterial nature1 procures perfection from the imperfect1 accurate subtilty from that which is neither accurate nor subtle1 and rekindles the light of ideas from the obscure and unreal objects of sense4 ,for where shall we find1 amongst sensible objects1 an indivisible nature1 such as that of a point1 or a line without the dimension of breadth1 or a superficies without depth1 or the ever constant proportion of sides1 and exact rectitude of angles8 ,for my part1 ,i cannot see where1 since all divisible natures are thus mixed and confused together1 nothing sincere1 nothing free from its contrary1 but things every where yielding to separation1 as well such as are removed by distance of place1 as those which are united together4 ,how then shall we obtain this durable essence for these immoveable natures from the ever fluctuating forms of sense8 ,for whatever derives its existence from moveable beings1 must of necessity be mutable and frail4 ,and how shall we gain this perfect accuracy for the stable species1 from the inaccurate and imperfect8 ,for whatever is the cause of a conception1 always immutable1 is itself much more stable than its effect4 ,we must therefore admit the soul to be the generator of these mathematical species and reasons4 ,but if she contains them in herself1 as first exemplars1 she gives them an essential being1 so that the generations are nothing else than propagations of species1 which had a prior subsistence in herself3 and thus we shall speak agreeably to the sentiments of ,plato1 and discover the true essence of mathematical entities4 ,but if the soul1 though she neither possesses nor received the mathematical reasons prior to the energies of sense1 yet fabricates this admirable immaterial building1 and generates this fair series of speculations2 how can she discern whether her productions are stable and constant1 or things which the winds may dissipate1 and phantoms rather than realities8 ,what standard can she apply as the measure of their truth8 ,or how1 since she is destitute of their essence1 can she generate such a variety of reasons8 ,for from such an hypothesis1 we make their subsistence fortuitous1 not tending to any scientific bound4 ,mathematical species are therefore the genuine offspring of the soul3 nor does she derive from sensible objects the definitions she frames1 but rather the first are propagated from the second2 they are the energies of soul1 which1 as it were1 pregnant with forms1 delivers her immaterial progeny into the dark and fluctuating regions of matter1 as evidences of the permanent duration of her species4 ,again1 if we collect mathematical reasons from externals1 why are not demonstrations composed from sensibles1 better than the demonstrations of universal and simple species8 ,for we say1 in order to the investigation of any thing sought1 that the principles and propositions1 should be allied to the conclusions4 ,if then1 particulars are the causes of universals1 and sensibles the sources of reasoning1 why does the boundary of demonstration always refer to that which is more universal1 and not to that which is partial and particular8 ,and how can we prove that the essence of intelligibles is more allied to demonstration than the essence of sensibles8 ,for thus they speak#gf3 his knowledge is not legitimate1 who demonstrates that the isosceles1 the equilateral1 or the scalene triangle1 have angles equal to two right2 but he possesses science1 properly so called1 who demonstrates this of every triangle simply1 or of triangle itself4 ,and again1 that universals1 for the purpose of demonstration1 are superior to particulars2 that demonstrations concern things more universal2 but that the principles from which demonstrations are composed1 have a priority of existence1 and a precedency in nature to singulars1 and are the causes of the propositions they prove4 ,it is very remote1 therefore1 from the nature of ,apodictical sciences1 that from converse with things of posterior origin1 and from the dark perceptions of sense1 they should falsely collect their indubitable propositions4 ,i add farther1 that they who affirm this1 make the soul of a baser nature than the material species themselves4 ,for if matter derives from nature beings essential1 and participating a high degree of entity and evidence2 but the soul1 by a posterior energy1 receives these from sensible objects1 and fashions in herself resemblances and images of posterior origin1 contemplating vile essences1 and abstracting from matter1 the forms inseparable from its nature2 do they not make the soul more obscure and indigent than matter itself8 ,for matter is the receptacle of forms materialized1 as the soul is of species immaterialized4 ,but in this case1 matter would be the place of primary beings1 and the soul of such as are secondary and subordinate3 matter and its forms obtaining the lead in being1 and existing as the sources of the subsistence of immaterial forms4 ,lastly1 the material forms would have an essential existence1 the others only an intentional denomination4 ,how then can the soul1 which is the first participant of intellect1 and an intellective essence1 and which derives from thence consummate knowledge1 and a plenitude of life1 become the receptacle of the most obscure species1 the lowest in the order of things1 and participating the most imperfect existence4 ,but this opinion1 which has been sufficiently exploded by others1 needs no farther confutation4 ,if then1 mathematical species do not subsist by material abstraction1 nor by a collection of those common properties inherent in individuals2 nor are at all1 in their origin1 posterior to sensibles1 nor derived in any manner from them3 it is necessary that the soul should either deduce them from herself1 or from intellect2 or lastly1 from herself and intellect united4 ,but if from herself alone1 ,whence do the images of intellectual species arise2 whence do they derive their middle nature1 linking1 as it were1 the divisible and indivisible essence together2 if they do not participate the fullness of entity from primary essences8 ,lastly1 how1 upon this hypothesis1 are the first exemplars1 paradigms1 or ideas1 which subsist in intellect1 the principles of universals8 ,but if they are derived from intellect alone into the soul1 how can the soul remain self-operative1 and self-motive1 if her inherent reasons flow from an external source1 and are regulated by its operations8 ,and in what respect does the soul differ from matter1 which is all things in mere dormant capacity1 but generates nothing appertaining to material species8 ,it remains1 therefore1 that the soul deduces these species from herself1 and intellect2 and that she is the absolute consummation of the forms which originate from intellectual exemplars1 but which are allotted from themselves a transition to permanent being4 ,the soul1 therefore1 is by no means to be compared to a smooth tablet1 void of all reasons2 but she is an ever-written tablet1 herself inscribing the characters in herself1 of which she derives an eternal plenitude from intellect4 ,for soul is a certain subordinate intellect1 revolving round an intellect prior to herself1 formed to its image1 and participating its divine irradiations4 ,if then1 this superior intellect is all things intellectually1 soul will all things animally2 if the first exists as the exemplar1 soul will be as its image2 if as contracted and united in itself1 soul as divisible and expanded4 ,and this is what ,plato understood1 when in his ,timus1 he composes the soul of the world from all things1 dividing her according to harmonical reasons1 and analogies2 assigning to her the first principles effective of figures1 ,i mean the right and circular line1 and giving an intellectual motion to her inherent circles4 ,all mathematical species1 therefore1 have a primary subsistence in the soul3 so that1 before sensible numbers1 there are to be found in her inmost recesses1 self-moving numbers2 vital figures1 prior to the apparent1 ideal proportions of harmony previous to concordant sounds2 and invisible orbs1 prior to the bodies which revolve in a circle4 ,so that soul is the prolific abundance of all these1 and is another ornament producing herself1 and produced from a proper principle1 filling herself with life1 and at the same time filled from the demiurgus of the universe1 is an incorporeal and indistant manner4 ,when1 therefore1 she produces and unfolds her latent reasons1 she then detects every science and virtue4 ,the essence of soul then consists in these species1 nor must we suppose her inherent numbers to be a multitude of units1 nor her archytipal ideas of divisible forms to be corporeal3 but we must conceive all these as subsisting ever vitally1 and intellectually1 as the exemplars of apparent numbers1 figures1 reasons and motions4 ,and here we must follow the doctrine of ,timus1 who derives the origin1 and consummates the fabric of the soul1 from mathematical forms1 and reposes in her nature the causes of every thing which exists4 ,for the seven bounding terms#gg1 comprehending the principles of all numbers1 lines1 planes and solids1 pre-exist in soul according to cause4 ,and again1 the principles of figures are placed in her essence1 according to a demiurgical power4 ,and lastly1 the first of all motions1 which embraces every other motion in its comprehensive ambit1 is co-existent with soul4 ,for the principle of every thing which is moved is a circle1 and the circular motion4 ,the mathematical reasons1 therefore1 which fully consummate the soul1 are essential1 and self-moving3 and the soul1 by her cogitative power1 diffusing1 propagating1 and evolving these1 from her profound recesses1 constitutes all the fair variety of mathematical sciences4 ,nor will she ever cease to generate1 and waken into energy1 succeeding species1 while she divests her indivisible reasons of their intellectual simplicity4 ,for she previously received all things1 after a primary manner2 and according to her infinite power1 from pre-existent principles1 deduces a beautiful series of various speculations4 ,,chap4 ,,vii4 ,what the ,employments and ,powers are of the ,mathematical ,science1 and how far they extend themselves in their ,energies4 ,but1 after contemplating the essence of mathematical forms1 it is necessary we should recur to that one master-science of these1 which we have shewn is prior to a multitude of others1 and that we should contemplate what its employment is1 what are its powers1 and how far it advances in its energies4 ,the employment1 therefore1 of the whole mathematical science1 possessing1 as we have before said1 the power of cogitation1 must not be placed so high as that of intelligence2 which is firmly seated in its own stable essence1 is perfect1 is contained by itself1 and in itself continually verges4 ,nor must it be situated so low as that of opinion and sense1 since these cognitions dwell upon external concerns1 energize upon them1 and do not possess the causes of the objects of their knowledge4 ,but the mathematical science1 receives its commencement1 indeed1 extrinsically from#gh recollection1 but ends in the most intimate reasons1 residing in the depths of the soul2 and is excited1 indeed1 from things posterior1 but arrives by gradual advances at the principal essence of forms4 ,nor is its energy immoveable1 like that of intelligence1 nor is it affected with local motion and alteration1 like sense1 but it revolves with a vital energy1 and runs through the ornament of incorporeal reasons1 sometimes advancing from principles to such things as are perfected by principles1 but at other times yielding in a retrograde progression from conclusions to their forming principles3 and sometimes proceeding from things previously known1 to such as are the subject of investigation3 but at other times1 from things placed in the question1 to such as precede in cognition4 ,besides1 it does not excel all inquisition1 as if it were perfect from itself1 like intellect1 nor is it perfected from others1 like sense1 but it proceeds by enquiry to invention1 and ascends from the imperfect to perfection4 ,but it likewise possesses twofold powers1 one kind of these deducing principles into multitude1 and generating the different paths of contemplation3 but the other endued with a power of collecting many transitions into proper suppositions4 ,for since it proposes to itself as principles1 as well unity1 and multitude1 as bound and infinite1 and such things as are subject to its comprehension1 are allotted a middle order1 between forms indivisible and every way divisible2 with great propriety 7,i think7 the gnostic powers of the whole science of these are essentially twofold4 ,one species indeed1 hastens to union1 and contracts the expansion of multitude3 but the other possesses a power of distinguishing things simple into such as are various1 more universals into more particulars1 and reasons digested in their principle1 into things secondary and multifariously multiplied from these principles4 ,for rising higher from its commencement it penetrates even to such things as are the perfections of sensible concerns1 is joined with nature1 and demonstrates many things together with natural science4 ,since ascending from inferiors1 it accedes in a certain respect proximate to intellectual knowledge1 and touches the contemplation of things primary and divine4 ,and hence1 in the limits which flow from its essence1 it produces the whole mechanic1 optic1 and catoptric speculation1 together with many other sciences which are inwoven and entangled with sensible concerns1 and which operate through their assistance4 ,besides1 in its ascensions from corporeal natures1 it derives intelligences indivisible and destitute of matter3 and with these it perfects its divisible apprehensions1 those cognitions which subsist in progressions1 and its own genera and forms3 it likewise indicates the truth respecting the gods themselves1 and in its peculiar treatises exhibits a contemplation of the things which are4 ,and thus much concerning the employment and powers of the ,mathematical ,science4 ,,chap4 ,,viii4 ,concerning the ,utility of the ,mathematical ,science4 ,but let us now consider the utility of this ,science1 which extends itself from the most principal to the last cognitions4 ,timus1 therefore1 calls the knowledge of the mathematical disciplines the path of erudition1 because1 indeed1 it has the same proportion to universal science1 and the first philosophy1 which learning has to virtue4 ,for this last frames our soul to a perfect life1 by the possession of worthy manners2 but the former prepares our cogitation1 and the divine eye of our soul to an elevation from the obscurity of sensible information4 ,hence1 ,socrates in the ,republic1 says1 8,that the eye of the soul1 which is darkened and buried by other studies1 can by the mathematical disciplines alone be invigorated1 and again excited to the contemplation of that which is1 and transferred from resemblances to real beings1 from an obscure light to that light which has the power of intelligence1 and from a cave1 and those bonds which exist in it as the authors of generation1 and from material impediments be able to rise to an incorporeal and indivisible essence4 ,for the beauty and order of mathematical reasons1 and the firmness and stability of the contemplations they afford1 conjoins us with intelligible objects1 and perfectly determines us in their essences2 which perpetually remain the same1 ever shining with divine beauty1 and preserving a mutual order without end4 ,but ,socrates1 in the ,phdrus1 delivers to us three characters who are elevated from sense1 because they fill up and accomplish the primary life of the soul1 i4 e4 the philosopher1 the lover1 and the musician4 ,but the beginning and path of elevation to the lover1 is a progression from apparent beauty1 using as excitations the middle forms of beautiful objects4 ,but to the musician1 who is allotted the third seat1 the way consists in a transition from sensible to invisible harmonies1 and to the reasons existing in these4 ,so that to the one1 sight is the instrument of reminiscence1 and to the other1 hearing4 ,but to him who is by nature a philosopher1 from whence and by what means is reminiscence the prelude of intellectual knowledge1 and an excitation to that which truly is1 and to truth itself8 ,for this character also1 on account of its imperfection1 requires a proper principle3 for it is allotted a natural virtue1 an imperfect eye1 and a degraded manner4 ,it must therefore be excited from itself2 and he who is of such a nature1 rejoices in that which is4 ,but to the philosopher1 says ,plotinus1 the mathematical disciplines must be exhibited1 that they may accustom him to an incorporeal nature1 and that afterwards using these as figures1 he may be led to dialectic reasons1 and to the contemplation of all the things which are4 ,and thus it is manifest1 from hence1 that the mathematics are of the greatest utility to philosophy4 ,but it is requisite that we should be more explicit1 and mention the several particulars to which they conduce1 and evince that they prepare the intellectual apprehensions of theology4 ,for whatever to imperfect natures appears difficult and arduous in obtaining the true knowledge of the gods1 the mathematical reasons render1 by their images1 credible1 manifest1 and certain4 ,thus1 in numbers1 they indicate the significations of super-essential properties1 but they evince the powers of intellectual figures1 in those figures which fall under cogitation4 ,hence it is1 that ,plato1 by mathematical forms teaches us many and admirable sentences concerning the gods1 and the philosophy of the ,pythagoreans1 using these as veils1 conceals from vulgar inspection the discipline of divine sentences4 ,for such is the whole of the ,sacred and ,divine ,discourse#gi1 that of ,philolaus in his ,bacchics1 and the universal method of the ,pythagoric narration concerning the ,gods4 ,but it especially refers to the contemplation of nature1 since it discloses the order of those reasons by which the universe is fabricated1 and that proportion which binds1 as ,timus says1 whatever the world contains1 in union and consent2 besides1 it conciliates in amity things mutually opposing each other1 and gives convenience and consent to things mutually disagreeing1 and exhibits to our view simple and primary elements1 from which the universe is composed1 on every side comprehended by commensurability and equality1 because it receives convenient figures in its proportions1 and numbers proper to every production1 and finds out their revolutions and renovations1 by which we are enabled to reason concerning the best origin1 and the contrary dissolution of particulars4 ,in consequence of this1 as it appears to me1 ,timus discloses the contemplation concerning the nature of the universe1 by mathematical names1 adorns the origin of the elements with numbers and figures1 referring to these their powers1 passions1 and energies2 and esteeming as well the acuteness as the obtuseness of angles1 the levity of sides1 or contrary powers1 and their multitude and paucity to be the cause of the all-various mutation of the elements4 ,but why may we not say1 that it profits much1 and in an admirable manner1 to that philosophy which is called ,politic1 as well by measuring the times of actions as affording the various revolutions of the universe1 and numbers convenient to things rising into being2 ,i mean the assimilating1 and authors of dissimilitude1 the prolific too and the perfect1 and the contraries to these2 together with orderly and elegant ministers of life1 and inelegance2 and finally1 such numbers as procure fertility and sterility4 ,which1 indeed1 the speech of the ,muses in the ,republic#hj evinces1 placing the universal ,geometric ,number as the author of better and more debased generations1 and as the cause of the indissoluble perseverance of good manners1 and of the mutation of the best ,republics into such as are remote from reason1 and are given to affections4 ,for it is sufficiently evident1 that it belongs to the whole mathematical discipline to deliver the science of this number which is called geometrical1 and not to one particular science1 such as arithmetic1 or geometry3 since the reasons or proportions of abundance and sterility1 permeate through all the mathematical disciplines4 ,again1 it is the means of our institution in moral philosophy which it brings to its ultimate perfection1 and gives order and an elegant life to our manners4 ,besides this1 it delivers to us figures1 and modulations and motions convenient to virtue1 by which the ,athenian guest wishes those to be instituted and perfected1 who are destined to pursue moral virtue from their early youth4 ,add too1 that it places before our view the reasons of virtues1 in one manner1 indeed1 in numbers1 in another in figures1 but differently in musical symphonies2 and lastly1 it indicates the excess and defect of vices1 by which we are enabled to moderate and adorn our manners4 ,hence it is1 that ,socrates1 in the ,gorgias1 accusing ,calicles of an inordinate and intemperate life1 says to him1 8,you neglect geometry and geometric equality30 but1 in the ,republic1 he finds out the proportion of tyrannic pleasure to a royal interval1 according to a plane and solid generation4 ,but we shall learn what great utility is derived to other sciences and arts from the mathematical science1 when we consider that it adds order and perfection to contemplative arts2 ,i mean rhetoric1 and all such as consist in discourse4 ,but it proposes to the poetic arts1 the reasons of poems in the place of an example1 because it presides over the measures existing in these4 ,but to the active arts it determines action and motion1 by its own abiding and immoveable forms4 ,for all arts1 as ,socrates says1 in the ,philebus1 require arithmetic1 mensuration1 and statics1 either in all1 or in some of their operations4 ,but all these are contained in the discourses of the mathematical science1 and are terminated according to their diversity4 ,for from this science the divisions of numbers1 and the variety of dimensions1 and the difference of weights are known4 ,the utility1 therefore1 of the whole mathematical science to philosophy itself1 and to other sciences and arts1 may be from hence known to intelligent hearers40 ,,chap4 ,,ix4 ,a ,solution of an ,objection raised by some against the ,utility of the ,mathematical ,sciences4 ,but some1 who are prone to contradiction through those who wish to subvert geometry1 endeavour to destroy the dignity of this science4 ,one part1 indeed1 depriving it of ornament and good1 because it does not discourse on these4 ,but another part#ha affirming that sensible experiments are more useful than the universal objects of its speculation2 ,i mean1 that ,geodesia 7for instance17 or the mensuration of the earth1 is preferable to geometry1 and vulgar arithmetic to that arithmetic which is conversant with theorems alone3 and that nautical astrology is more useful than that which teaches universally1 abstracted from any application to sensible concerns4 ,for we are not1 say they1 made rich by our knowledge of riches1 but by using them2 nor are we happy by the merely understanding felicity1 but by living happily4 ,hence we must confess that those mathematical sciences1 which are conversant with cognition1 do not profit human life1 and confer to action1 but those only which are engaged in exercise4 ,for those who are ignorant of the reasons of things1 but are exercised in particular and sensible experiments1 are in every respect more excellent1 for the purposes of human life1 than those who are employed in contemplation alone4 ,against objections then1 of this kind1 we shall reply1 by shewing the beauty of the mathematical disciplines from those arguments by which ,aristotle endeavours to persuade us4 ,we must therefore confess that there are three things which especially cause beauty1 both in bodies and souls2 ,i mean1 order1 convenience1 and determination4 ,since corporeal baseness1 indeed1 arises from material inordination1 deformity1 and inconvenience1 and from the dominion of the indefinite in the composite body4 ,but the baseness of the soul originates from its irrationality1 and inordinate motion1 and from its being in a state of discord with reason1 and not receiving from thence its proper limitation4 ,hence1 beauty exists even in contraries1 by means of order1 convenience and determination4 ,but we may behold these in a more eminent degree in the mathematical science2 order1 indeed1 in the perpetual exhibition of things posterior and more various1 from such as are primary and more simple2 for things subsequent are always annexed to their precedents1 the latter ranking as principles1 and the former as the first suppositions of things consequent3 but convenience is evinced in the mutual consonance of things demonstrated1 and in the relation of all of them to intellect1 since intellect is the common measure of all science1 from which it receives its principles1 and to which it converts the learner3 but determination is perceived in its perpetually abiding and immoveable reasons1 for the objects of its knowledge are not1 at times1 subject to variation1 like those of opinion and sense1 but present themselves for ever the same1 and are bounded by intellectual forms4 ,if such then1 are the principal requisites of beauty1 it is evident1 that in these sciences that illustrious ornament and gracefulness is found4 ,for how is it possible this should not be the case with a science receiving a supernal illumination from intellect1 to which it continually advances1 hastening to transfer us from the obscure light of sensible information8 ,with respect to the second objection1 we think it proper to judge of its utility1 without regarding the conveniencies and necessities of human life4 ,for otherwise1 we must confess that contemplative virtue is also useless1 which separates itself from human concerns1 which it is very little desirous to look down upon and understand4 ,indeed ,socrates1 in the ,thetetus1 affirming this concerning noblemen endued with the prophetic power1 says1 8that it withdraws them from all regard to human life1 and raises their thoughts1 properly liberated1 from all necessity and use1 to the very summit of all true being40 ,the mathematical science1 therefore1 must be considered as desirable for its own sake1 and for the contemplation it affords1 and not on account of the utility it administers to human concerns4 ,but if it is necessary to refer the utility it produces to something different from itself1 it must be referred to intellectual knowledge4 ,for it leads us to this1 and prepares the eye of the soul for the knowledge of universals1 removing and obliterating the impediments arising from the senses1 and from corporeal involution4 ,as therefore we call the whole of purgative virtue useful1 or the contrary1 not regarding the use of the sensible life1 but of that which is contemplative1 so indeed it is requisite to refer the end of mathematics to intellect1 and universal wisdom4 ,hence its energy is worthy our study1 both on its own account1 and on account of an intellectual life4 ,but it appears1 as ,aristotle#hb says1 that this science is desirable of itself to its votaries1 because though no reward is proposed to its enquirers1 yet the mathematical contemplation receives1 in a small time1 an abundant increase4 ,besides1 this is farther evident from hence1 that all men are willingly employed in its pursuit1 and wish to dwell on its speculations1 omitting every other concern2 even those who have1 with their lips1 as it were1 but just touched its utility4 ,and hence it follows1 that they who despise the knowledge of the mathematical disciplines1 have very little tasted of the pleasures they contain4 ,the mathematics1 therefore1 are not to be despised because their speculative parts do not immediately confer to human utility1 7for the ultimate limits of its progressions1 and whatever operates with matter1 consider a use of this kind27 but on the contrary we should admire its immateriality1 and the good which it contains1 considered by itself alone4 ,for when mankind were entirely disengaged from the care of necessary concerns1 they converted themselves to the investigation of the mathematical disciplines2 and this1 indeed1 with the greatest propriety4 ,since affairs familiar to human life in its most imperfect state1 and which are immediately connected with its origin1 first of all employed the studies of mankind3 but1 in the second place1 those concerns succeeded which separate the soul from generation1 and restore its memory of that which ,,is4 ,after this manner1 then1 we are engaged in necessaries1 before things honourable for their own sakes1 on account of their intrinsic dignity and worth2 and in things related to sense1 before such as are apprehended by the nobler energies of mind4 ,for every origin and life of the soul which is converted into herself1 is naturally adapted to proceed from the imperfect to the perfect4 ,and thus much against those who despise the mathematical science40 ,,chap4 ,x4 ,a ,solution of another ,objection of certain ,platonists1 against the ,utility of the ,mathematical ,sciences4 ,but1 perhaps1 some of our own family will here rise up against us1 and1 proposing ,plato as a witness1 will endeavour to provoke ruder understandings into a contemptuous disregard of the mathematical disciplines4 ,for they will say1 that this philosopher entirely excludes 7in his ,republic7 the mathematical knowledge from the choir of the sciences1 and that he accuses it as being ignorant of its own principles1 that its very principle is to itself unknown1 and its ends and mediums composed from things of which it is ignorant4 ,to these objections they may likewise add whatever other reproaches are there urged by ,socrates against this contemplation4 ,in answer then1 to the objections of our friends1 we shall recall into their memory1 that ,plato himself perspicuously asserts the mathematical science to be the purgation of the soul1 and that it is endued with a power of leading it on high2 because1 like the ,homeric ,minerva1 it removes the darkness of a sensible nature from the intellectual light of thought1 which is better worth saving than ten thousand corporeal eyes1 and which not only participates of a mercurial gift1 7preserving us from the incantations and delusions of this material abode1 which is similar to the fascinating realms of ,circe17 but also of the more divine arts of ,minerva4 ,he likewise every where calls it by the name of science1 and asserts that it is the cause of the greatest felicity to those who are exercised in its contemplation4 ,but ,i will briefly explain why1 in the ,republic he takes from it the surname of science3 for my present discourse is addressed to the learned4 ,plato1 indeed1 in most places1 calls all the knowledge 7as ,i may say7 of universals by the name of science1 opposing it in a division to sense which apprehends only particulars1 whether such a mode of cognition is accomplished by art or experience4 ,and in this sense1 as it appears to me in the ,civil ,dialogue1 and in the ,sophista1 he seems to use the name of science2 placing likewise the illustrious ,sophistic science1 which ,socrates in the ,gorgias1 says1 is a certain experience3 also1 the adulatory1 and many others1 which are experiences1 but not true sciences4 ,but again1 dividing this knowledge of universals into that which knows causes1 and into that which understands without a cause1 he thinks that the one should be called science1 but the other experience4 ,and hence1 to arts he sometimes attributes the name of science1 but to experience never4 ,for how 7says he in the ,banquet7 can a thing which possesses no reason be science8 ,all knowledge1 therefore1 which contains the reason and cause of the things known1 is a certain science4 ,again1 therefore1 he divides this science which is endued with a power from the cause of knowing1 by the peculiarity of its subjects1 and he places one1 conjectural of things divisible2 but the other of such as subsist by themselves1 and are ever knowable after the same manner4 ,and according to this division he separates from science1 medicine1 and every faculty which is conversant with material concerns4 ,but he calls mathematical knowledge1 and whatever possesses a power of contemplating eternal objects1 by the name of science4 ,lastly1 dividing this science1 which we distinguished from arts1 he considers one part as void of supposition2 but the other as flowing from supposition4 ,and that the one which is void of supposition1 has a power of knowing universals3 that it rises to good1 and the supreme cause of all2 and that it considers good as the end of its elevation3 but that the other1 which previously fabricates for itself definite and determinate principles1 from which it evinces things consequent to such principles1 does not tend to the principle1 but to the conclusion4 ,and hence he asserts1 that mathematical knowledge1 because it makes use of supposition1 falls short of that science which is without supposition1 and is perfect4 ,for there is one true science1 by means of which we are disposed to know all the things which are1 and from which also principles emerge to all sciences2 to some1 indeed1 constituted more proximately1 but to others more remotely4 ,we must not say1 therefore1 that ,plato expels mathematical knowledge from the number of the sciences1 but that he asserts it to be the second from that one science1 which possesses the supreme seat of all3 nor must we affirm1 that he accuses it as ignorant of its own principles1 but that receiving these from the master science dialectic1 and possessing them without any demonstration1 it demonstrates from these its consequent propositions4 ,for1 indeed1 he sometimes allows the soul1 which is constituted from mathematical reasons1 to be the principle of motion3 and sometimes he affirms1 that it receives its motion from genera which are subject to intelligence4 ,and these variations accord among themselves4 ,for to such things as are moved by another1 the soul is a certain cause of motion1 but it is not the cause of every motion4 ,after the same manner1 the mathematical science is indeed the second from the first of all sciences1 and1 with reference to it1 imperfect3 but it is1 nevertheless1 a science1 not as being free from supposition1 but as knowing the peculiar reasons resident in the soul1 and as bringing the causes of conclusions1 and containing the reason of such things as are subject to its knowledge4 ,and thus much for the opinion of ,plato respecting mathematics4 ,,chap4 ,,xi4 ,but let us now consider what are the things which may be required of a mathematician1 and how any one may rightly judge concerning his distinguishing peculiarities4 ,for#hc ,aristotle indeed1 says1 that he who is simply learned in all disciplines1 is adapted to judge of all3 but that he who is alone skilled in the mathematical sciences1 can alone determine concerning the magnitude of reasons inherent in these4 ,it is requisite1 therefore1 that we should previously assume the terms of judging1 and that we should know1 in the first place1 in what things it is proper to demonstrate generally1 and in what to regard the peculiarities of singulars4 ,for many of the same properties reside in things differing in species1 as two right angles in all triangles3 but many have indeed the same predicament1 yet differ in their individuals in a common species1 as similitude in figures and numbers4 ,but one demonstration is not to be sought for by the mathematician in these1 for the principles of figures and numbers are not the same1 but differ in their subject genus4 ,and if the essential accident is one1 the demonstration will also be one#hd3 for the possession of two right angles is the same in all triangles1 and that general something to which this pertains is the same in all1 ,i mean triangle1 and a triangular reason4 ,in the same manner1 likewise1 the possession of external angles to four right ones1 not only pertains to triangles1 but also to all right-lined figures2 and the demonstration1 so far as they are right-lined1 agrees in all4 ,for every reason brings with it1 at the same time1 a certain property and passion1 of which all participate through that reason1 whether triangular1 or rectilinear1 or universally figure4 ,but the second limit by which a mathematician is to be judged1 is1 if he demonstrates according to his subject-matter1 and renders necessary reasons1 and such as cannot be confuted1 but are at the same time neither probable1 nor replenished with a similitude of truth4 ,for1 says ,aristotle1 it is just the same to require demonstrations from a rhetorician1 and to assent to a mathematician disputing probably2 since every one1 endued with science and art1 ought to render reasons adapted to the subjects of his investigation4 ,in like manner also1 ,plato in the ,timus1 requires credible reasons of the natural philosopher1 as one who is employed in the resemblances of truth3 but of him who discourses concerning intelligibles1 and a stable essence1 he demands reasons which can neither be confuted nor moved4 ,for subjects every where cause a difference in sciences and arts1 since1 if some of them are immoveable1 others are conversant with motion2 and some are more simple1 but others more composite2 and some are intelligibles1 but others sensibles4 ,hence we must not require the same certainty from every part of the mathematical science4 ,for if one part1 after a manner1 borders upon sensibles1 but another part is the knowledge of intelligible subjects1 they cannot both be equally certain1 but one must inherit a higher degree of evidence than the other4 ,and hence it is1 that we call arithmetic more certain than the science of harmony4 ,nor must we think it just that mathematics and other sciences should use the same demonstrations2 for their subjects afford them no small variety4 ,in the third place1 we must affirm1 that he who rightly judges mathematical reasons1 must consider sameness and difference1 what subsists by itself1 and what is accidental1 what proportion is1 and every consideration of a similar kind4 ,for almost all errors of this sort happen to those who think they demonstrate mathematically1 when at the same time they by no means demonstrate1 since they either demonstrate the same thing as if different in each species1 or that which is different as if it were the same3 or when they regard that which is accidental1 as if it were an essential property2 or that which subsists by itself1 as if it were accidental4 ,for instance1 when they endeavour to demonstrate that the circumference of a circle is more beautiful than a right line1 or an equilateral than an isosceles triangle4 ,for the determination of these does not belong to the mathematician1 but to the first philosopher alone4 ,lastly1 in the fourth place1 we must affirm1 that since the mathematical science obtains a middle situation between intelligibles and sensibles1 and exhibits in itself many images of divine concerns1 and many exemplars of natural reasons1 we may behold in it three kinds of demonstration#he1 one approaching nearer to intellect1 the second more accommodated to cogitation1 and the third bordering on opinion4 ,for it is requisite that demonstrations should differ according to the varieties of problems1 and receive a division correspondent to the genera of beings1 since the mathematical science is connected with all these1 and adapts its reasons to the universality of things4 ,and thus much for a discussion of the subject proposed4 ,,chap4 ,,xii4 ,what and how many the ,species of the whole ,mathematical ,science are1 according to the ,opinion of the ,pythagoreans4 ,but after these considerations1 it is requisite to determine concerning the parts of the mathematical science1 what1 and how many they are4 ,for it is just1 after speculating its whole and entire genus1 to consider the differences of its more particular sciences1 according to their species4 ,the ,pythagoreans#hf1 therefore1 thought that the whole mathematical science should receive a fourfold distribution1 attributing one of its parts to the how-many1 but the other to the how-much2 and they assigned to each of these parts a twofold division4 ,for they said1 that discrete quantity1 or the how-many1 either subsists by itself1 or must be considered with relation to some other2 but that continued quantity1 or the how-much1 is either stable or in motion4 ,hence they affirmed1 that arithmetic contemplates that discrete quantity which subsists by itself1 but music that which is related to another2 and that geometry considers continued quantity so far as it is immoveable2 but spherics contemplates continued quantity as moving from itself1 in consequence of its union with a self-motive nature4 ,they affirmed besides1 that these two sciences1 discrete and continued quantity1 did not consider either magnitude or multitude absolutely1 but that alone which in each of these is definite from the participation of bound4 ,for sciences alone speculate the definite1 rejecting as vain the comprehension of infinite quantity4 ,but when these wise men assigned this distribution1 we must not suppose they understood that discrete quantity which is found in sensible natures1 nor that continued quantity which subsists about the fluctuating order of bodies4 ,for1 ,i think1 the contemplation of these pertains to the natural and not to the mathematical science4 ,but because the demiurgus of the universe1 employed the union1 division1 and identity of general natures1 together with difference1 station1 and motion1 for the purpose of completing the essence of the soul1 and composed it from these genera1 as ,timus informs us1 we must affirm1 that cogitation1 abiding according to its diversity1 its division of reasons1 and its multitude1 and understanding itself to be both one and many1 proposes indeed to itself1 and produces numbers1 together with an arithmetical knowledge of these3 but it provides for itself music according to an union of its multitude1 and a communication and junction with itself2 and hence it is that arithmetic excels music in antiquity2 since1 according to the narration of ,plato1 the demiurgus first divided the soul1 and afterwards collected it in harmonical proportions4 ,again1 thought establishing its energy according to the stability which it contains1 draws from its inmost retreats geometry1 together with one essential figure1 and the demiurgical principles of all figures#hg3 but1 according to its inherent motion1 it produces the spherical science4 ,for it is moved also by circles1 but abides perpetually the same from the causes of circles4 ,hence1 likewise1 geometry precedes spherics1 in the same manner as station is prior to motion4 ,but because cogitation itself produces these sciences1 not by looking back upon its convolution of forms1 endued with an infinite power1 but upon the inclosure of bound according to its definite genera2 hence they say1 that the mathematical sciences take away infinite from multitude and magnitude1 and are only conversant about finite quantity4 ,indeed1 intellect has placed in cogitation all the principles both of multitude and magnitude4 ,for since it wholly consists1 with reference to itself1 of similar parts1 and is one and indivisible1 and again divisible1 educing the ornament of forms1 it participates of bound and infinite1 from intelligible essences themselves4 ,but it understands1 indeed1 from its participation of bound1 and generates vital energies1 and various reasons from the nature of infinite4 ,the intellections1 therefore1 of thought1 constitute these sciences according to the bound which they contain1 and not according to an infinity of life2 since they bring with them an image of intellect1 but not of life4 ,such then is the opinion of the ,pythagoreans1 and the division of the four mathematical sciences4 ,,chap4 ,,xiii4 ,another ,division of the ,mathematical ,science1 according to ,geminus4 ,again1 some think 7among whom is ,geminus7 that the mathematical science is to be divided in a different manner from the preceding4 ,and they consider that one of its parts is conversant with intelligibles only1 but the other with sensibles1 upon which it borders2 denominating as intelligibles whatever inspections the soul rouses into energy by herself1 when separating herself from material forms4 ,and of that which is conversant with intelligibles they establish two1 by far the first and most principal parts1 arithmetic and geometry3 but of that which unfolds its office and employment in sensibles1 they appoint six parts1 mechanics1 astrology1 optics1 geodsia1 canonics1 and logistics1 or the art of reckoning4 ,but they do not think that the military art1 or tactics1 should be called any one part of mathematics1 according to the opinion of some#hh2 but they consider it as using at one time the art of reckoning1 as in the numbering of legions2 but at another time geodsia1 as in dividing and measuring the spaces filled by a field of camps4 ,as1 say they1 neither the art of writing1 nor the art of healing1 are any part of mathematics1 though frequently both the historian and physician use mathematical theorems4 ,this is the case with historians indeed1 when relating the situation of climates1 or collecting the magnitudes and dimensions of cities1 or their compass and circuit3 but with physicians1 when elucidating by ways of this kind1 many things in their art4 ,for ,hippocrates himself shews the utility derived to medicine from astrology1 and almost all who speak of opportune times and places4 ,by the same reason he also1 who accommodates his work to tactics1 uses indeed mathematical theorems1 yet is not on this account a mathematician1 although he is sometimes willing that a numerous camp should exhibit a very small multitude1 and forms his army according to a circular figure2 but sometimes in a quadrangular1 quinquangular1 or some other multangular figure1 when he desires it to appear numerous4 ,but since these are the species of the whole mathematical science1 geometry is again divided into the contemplation of planes1 and the dimension of solids1 which is called stereometry4 ,for there is not any peculiar treatise about points and lines1 because no figure can be produced from these without planes or solids4 ,for geometry treats of nothing else in every one of its parts1 than that it may constitute either planes or solids3 or that when constituted1 it may compare and divide them among themselves4 ,in like manner1 arithmetic is distributed into the contemplation of linear1 plane1 and solid numbers4 ,for it considers the species of numbers separate from sensible connections1 proceeding from unity1 and the origin of plane numbers2 ,i mean of the similar1 dissimilar1 and solid1 even to the third increase4 ,but geodsia1 and the art of reckoning1 are divided similarly to arithmetic and geometry1 as they do not discourse concerning intelligible numbers or figures1 but of such as are sensible alone4 ,for neither is it the office of geodesia to measure the cylinder or the cone1 but material masses as if they were cones1 and wells as if they were cylinders4 ,neither does it accomplish this purpose by intelligible right lines1 but by such as are sensible1 sometimes indeed by a more certain means1 as by the solar rays3 but at other times by grosser ones1 as by a line and perpendicular4 ,in like manner1 the reckoner does not survey the passions of numbers by themselves1 but as they are resident in sensible objects4 ,from whence he also imposes a name upon these derived from the things which he reckons1 calling them 1 4 ,besides this1 he does not1 admit of any least1 like the arithmetician1 who receives that minimum1 as a genus of relation4 ,for some one man is considered by him as the measure of the whole multitude of men1 as unity also is the common measure of all numbers4 ,again1 optics and canonics are produced from geometry and arithmetic4 ,and optics uses the visual rays which are constituted by the rays of the eyes1 as lines and angles4 ,but it is divided into that which is properly called optics 7because it renders the cause of these appearances1 which are accustomed to present themselves to us different from their reality1 on account of the different situations and distances of visible objects1 as the coincidence of parallel lines1 or the appearance of quadrangles as if they were circles72 and into universal catoptrics1 which is conversant about various and manifold refractions1 and is connected with imaginative or conjectural knowledge3 as also into that which is called sciography#hi1 or the delineation of shadows1 which shews how appearances in images may seem neither inelegant nor deformed1 on account of the distances and altitudes of the things designed4 ,but canonics 7music7 or the regular art1 considers the apparent reasons of harmonies1 finding out the sections of rules1 every where using the assistance of sense1 and1 as ,plato says1 seeming to prefer the testimony of the ears to intellect itself4 ,but to the parts we have hitherto enumerated1 mechanics must he added1 as it is a certain part of the whole science1 and of the knowledge of sensible objects1 and of things united with matter4 ,but under this exists the art effective of instruments1 which is called 7o7 ,i mean of those instruments proper for the purposes of war3 such1 indeed1 as ,archimedes is reported to have constructed1 resisting the besiegers of sea and land2 and that which is effective of miracles1 and which is called 747 ,one part of this constructs with the greatest artifice pneumatic engines1 such as ,ctesibius and ,heron fabricated3 but another operates with weights1 the motion of which is reckoned to be the cause of inequilibrity2 but their station of equilibrity1 as ,timus also has determined3 and again1 another part imitates animate foldings and motions by strings and ropes4 ,again1 under mechanics is placed the knowledge of equilibriums1 and of such instruments as are called centroponderants3 also 77 or the art effective of spheres1 imitating the celestial revolutions1 such as ,archimedes fabricated2 and lastly1 every thing endued with a power of moving matter4 ,but the last of all is astrology1 which treats of the mundane motions1 of the magnitudes of the celestial bodies1 their figures and illuminations1 their distances from the earth1 and every thing of this kind2 assuming many things indeed to itself from sense1 but communicating much with the natural speculation4 ,one part of this is gnomonics1 which is exercised in settling the dimension of hororary gnomons3 but the other is metheoroscopics1 which finds out the differences of elevations1 and the distances of the stars1 and also teaches many other and various astrological theorems4 ,the third part is dioptrics1 which ascertains by dioptric instruments of this kind the distances of the sun and moon1 and of the five other stars4 ,and such is the account of the parts of the mathematical science1 delivered by the ancients1 and transmitted to our memory by the informing hand of time4 ,,chap4 ,,xiv4 ,how ,dialectic is the ,top of the ,mathematical ,sciences1 and what their ,conjunction is1 according to ,plato4 ,let us again consider after what manner ,plato1 in his ,republic1 calls dialectic the top of the mathematical disciplines2 and what their conjunction is1 according to the tradition of the author of the ,epinomis#ij4 ,and in order to this we must assert1 that as intellect is superior to cogitation1 supplying it with supernal principles1 and from itself giving perfection to cogitation2 in the same manner dialectic also1 being the purest part of philosophy1 excels in simplicity the mathematical disciplines1 to which it is proximate1 and with which it is conjoined4 ,indeed it embraces the complete circle of these sciences1 to which it elevates from itself various energies1 endued with a power of causing perfection1 judgment1 and intelligence4 ,and these energies consist in resolving1 dividing1 defining1 and demonstrating2 by which mathematics itself1 receiving assistance and perfection1 invents some things by resolution1 but others by composition3 and some things it explains by division1 others by definition3 but collects other subjects of its investigation by demonstration2 accommodating1 indeed1 these ways to its subjects1 but using each of them for the purpose of beholding its middle enquiries4 ,from whence indeed1 both the resolutions1 definitions1 divisions1 and demonstrations which it contains1 are peculiar1 and adapted to its nature1 and revolve according to the mode of mathematical cognition4 ,not undeservedly1 therefore1 is dialectic the vertex as it were1 and summit of mathematics4 ,since it perfects all which mathematics contains of intelligence2 renders its certainty free from reprehension1 preserves the stability of its immovable essence1 and refers what it contains destitute of matter and pure to the simplicity of intellect1 and a nature separated from material connections4 ,besides1 it distinguishes the first principles of these sciences1 by definitions3 exhibits the separations of genera and forms contained under the genera themselves3 and besides this1 teaches the compositions1 which1 from principles1 produce things consequent to principles3 and the resolutions which rise and mount up to things first1 and to principles themselves4 ,but with respect to what remains1 proportion itself is not to be considered 7as ,eratosthenes thought it was7 as the conjunction of the mathematical disciplines4 ,since proportion is said to be1 and indeed is one of those things common to the mathematics4 ,but in short1 many other things besides proportion regard all the mathematical disciplines1 which are essentially inherent in the common nature of the mathematics4 ,but as it appears to me1 we should say1 that there is one proximate conjunction of these1 and of the whole mathematical science1 which especially embraces in itself1 in a more simple manner1 the principles of all sciences2 which considers their community and difference2 teaches whatever is found in these the same2 together with what things are inherent in a many1 and what in a few4 ,so that to those who aptly learn there is a reversion from many other sciences to this alone#ia4 ,but1 dialectic is a conjunction of the mathematical disciplines superior to the preceding2 which ,plato1 as ,i have already observed1 calls in his ,republic their vertex3 for1 indeed1 it perfects the whole of mathematics1 brings it back to intellect by its powers1 shews it to be a true science1 and causes it to be certain and obnoxious to no reproof4 ,but1 intellect obtains the third order between these conjunctions1 which comprehends in itself uniformly all the dialectic powers1 contracts their variety by its simplicity1 their partition by its indivisible knowledge1 and their multitude by its occult union4 ,hence1 intellect itself congregates indeed the involutions and deviations of the dialectic paths1 into an intelligible essence1 but it collects supernally all the progression of mathematical discourses3 and it is the best end both of the elevating power of the soul1 and of the energy consisting in cognition4 ,and such are the sentiments declared by me on the present enquiry4 ,,chap4 ,,xv4 ,from whence the ,name ,mathematics originated4 ,again1 from whence shall we say this name of mathematics1 and mathematical disciplines1 was assigned by the ancients1 and what apt reason can we render of its position8 ,indeed1 it appears to me1 that such an appellation of a science which respects cogitative reasons1 was not1 like most names1 invented by indifferent persons1 but 7as the truth of the case is1 and according to report7 by the ,pythagoreans alone4 ,and this1 when they perceived1 that whatever is called mathesis or discipline1 is nothing more than reminiscence2 which does not approach the soul extrinsically1 like the images which rising from sensible objects are formed in the phantasy3 nor is it adventitious and foreign1 like the knowledge consisting in opinion1 but it is excited1 indeed1 from apparent objects1 and is perfected within1 by thought intimately converted to itself4 ,and when they likewise perceived that though reminiscence might be shewn from many particulars1 yet it was evinced in a more eminent manner 7as ,plato also says#ib7 from the mathematical disciplines4 ,for if any one1 says he1 is led into the descriptions1 he will there easily prove that discipline is reminiscence4 ,from whence ,socrates also1 in the ,meno1 shews by this method of arguing1 that learning is nothing else than the soul's recollection of her inherent reasons4 ,and this1 because that which recollects1 is alone the cogitative part of the soul2 but this perfects her essence in the reasons of the mathematical disciplines1 the sciences of which she previously received into herself1 though she does not always energize on their fair variety4 ,indeed1 she contains them all essentially and occultly2 but she produces each of them when she is freed from the impediments originating from sense4 ,for sense unites her with divisible objects3 the phantasy fills her with forming motions1 and appetite bends her to an indulgent and luxurious life4 ,but every thing divisible is an obstacle to our self-conversion4 ,and whatever invests with form1 disturbs and offends that knowledge which is destitute of form4 ,and whatever is obnoxious to perturbations is an impediment to that energy which is unimpaired by affections4 ,when1 therefore1 we have moved all these from the cogitative power1 then shall we be able to understand by thought itself1 the reasons which thought contains3 then shall we become scientific in energy2 and unfold our essential knowledge4 ,but whilst we are captive and bound1 and winking with the eye of the soul1 we cannot by any means attain to a perfection convenient to our nature4 ,such then is mathesis or discipline3 a reminiscence of the eternal reasons contained in the soul4 ,and the mathematical or disciplinative science is on this account particularly denominated that knowledge which especially confers to our reminiscence of these essential reasons4 ,hence1 the business and office of this science#ic1 is apparent from its name4 ,for its duty is to move the inherent knowledge of the soul2 to awaken its intelligence2 to purify its cogitation2 to call forth its essential forms from their dormant retreats2 to remove that oblivion and ignorance1 which are congenial with our birth2 and to dissolve the bonds arising from our union with an irrational nature4 ,it plainly leads us to a similitude of that divinity who presides over this science1 who manifests intellectual gifts1 and fills the universe with divine reasons2 who elevates souls to intellect1 wakens them as from a profound sleep1 converts them by enquiry to themselves2 and by a certain obstetric art1 and invention of pure intellect1 brings them to a blessed life4 ,to whom indeed1 dedicating the present work1 we here conclude our contemplation of the mathematical science4 ,,book ,,ii4 ,,chap4 ,i4 ,what ,part ,geometry is of ,mathematics1 and what the ,matter is of which it consists4 ,in the preceding discourses we have considered those common properties which respect the whole of the mathematical science2 and this we have done agreeable to the doctrine of ,plato2 at the same time collecting such particulars as pertain to our present design4 ,but consequent to this it is requisite that we should discourse on geometry itself1 and on the proposed institution of the elements1 for the sake of which we have undertaken the whole of the present work4 ,that geometry then1 is a part of the whole of mathematics1 and that it obtains the second place after arithmetic1 since it is perfected and bounded by this1 7for whatever in geometry may be expressed and known1 is determined by arithmetical reasons7 has been asserted by the ancients1 and requires no long discussion in the present enquiry4 ,but we also may be able to relate our opinion on this particular1 if we consider what place1 and what essence its subject matter#id is allotted among the universality of things4 ,for from a proper survey of this1 the power of the science which knows this subject matter1 the utility arising from it1 and the good acquired by its learners1 will immediately appear4 ,indeed1 some one may doubt in what genus of things he ought to place geometrical matter1 so as not to deviate from the truth it contains4 ,for if the figures concerning which geometry discourses1 exist in sensible natures1 and cannot be separated from the dark receptacle of matter2 how can we assert that geometry frees us from sensible objects1 that it brings us to an incorporeal essence1 that it accustoms us to an inspection of intelligibles1 and prepares us for intellectual energy8 ,where shall we ever survey among sensible objects a point without parts1 or a line destitute of breadth1 or a superficies without profundity1 or the equality of lines from the centre to the circumference2 or the multangles1 and all the figures of many bases1 concerning which geometry informs us8 ,lastly1 after what manner can the reasons of such a science remain free from all possible confutation2 since1 indeed1 sensible forms and figures are susceptive of the more and the less1 are all moveable and mutable1 and are full of material variety2 among which equality subsists mixt and confused with its contrary inequality1 and into which things without parts have proceeded into partition1 and interval1 darkened with the shades of matter1 and lost in its infinite folds8 ,but if the subjects of geometry are removed from matter1 are pure forms1 and are separated from sensible objects3 they will be all of them1 without doubt1 void of parts1 incorporeal1 and destitute of magnitude4 ,for extension1 tumor1 and interval1 approach to forms1 on account of the material receptacle in which they are involved1 and which receives things destitute of parts1 distributed into parts2 things void of dimension1 extended into dimension2 and immoveable natures accompanied with motion4 ,how then1 if this is the case1 shall we cut a right line1 triangle1 and circle8 ,how can we speak of the diversities of angles1 and the increments and decrements of triangular and quadrangular figures8 ,or how exhibit the contacts of circles or right lines8 ,for all these evince that the geometric matter consists of parts1 and does not reside among indivisible reasons4 ,such then are the doubts concerning the matter of geometry1 to which we may add1 that ,plato considers the forms of geometry as placed in cogitation2 and grants1 that we advance from sensibles to forms of this kind1 and that we rise from sensibles to intellect1 though 7as we have previously observed7 the reasons subsisting in cogitation are indivisible1 are separated by no interval1 and subsist according to the peculiarity of the soul4 ,but if reasons are to be rendered agreeable to things themselves1 and to the doctrine of ,plato1 the following division must be adopted4 #ie;,every universal1 and one thing containing many1 is either naturally disposed to be thought of in particulars1 or to appear such1 because it possesses its existence in these2 is inseparable from them2 is disposed and distributed in them2 and together with these is either moved1 or firmly and immoveably abides4 ,or it is adapted to subsist prior to many1 and to possess a power of generating multitude1 affording to many things images from itself1 being furnished with a nature destitute of parts1 from the essences which it participates1 and raising various participations to secondary natures3 or it is disposed to be formed by thought1 from the many1 to possess a generating existence1 and to reside in the last place in the many4 ,for1 according to these three modes of subsistence1 we shall find1 ,i think1 that some subsist before the many1 others in the many1 and others from the relation and predication which they possess to these4 ,but1 that ,i may absolve all in one word1 universal forms being threefold1 we shall consider the differences of that form which many participate1 which exists in many1 and fills particular natures according to its subject matter4 ,besides this1 establishing a twofold order of participants1 one subsisting in sensible objects1 but the other in the phantasy1 7since matter is twofold2 one indeed1 of things united with sense1 but the other of such as fall under the inspection of phantasy1 as ,aristotle asserts1 in a certain place#if7 we must allow that the universal1 which is distributed in the many1 is likewise twofold4 ,the one1 indeed1 sensible1 as being that which sensible objects participate2 but the other imaginative1 as that which subsists in the many of the phantasy4 ,for the phantasy1 on account of its forming motion1 and because it subsists with1 and in body1 always receives impressions which are both divided and figured4 ,so that whatever is known by it1 is allotted a correspondent existence3 on which account1 ,aristotle#ig does not hesitate to call it passive intellect4 ,but if it is intellect1 why is it not impassive1 and destitute of matter8 ,and if it operates with passion1 how can it with propriety be called intellect8 ,for impassivity1 indeed1 properly belongs to intellect and an intelligent nature3 but passivity is very remote from such an essence4 ,but 7unless ,i am deceived7 ,aristotle being willing to explain its middle nature between cognitions the most primary1 and such as are the last1 calls it at the same time intellect1 because similar to primary cognitions1 and passive from that alliance which it possesses with such as are posterior4 ,for first cognitions are indeed destitute of figures and forms2 comprehending in themselves1 intelligible natures1 energizing about themselves1 united with the objects of knowledge1 and free from all extrinsical impression and passion4 ,but last cognitions exercise themselves through the medium of instruments1 are rather passions than energies1 admit extrinsical knowledge1 and move themselves together with their various subjects4 ,for such 7says ,plato7 are the sensations which arise from violent passions4 ,but the phantasy1 obtaining a middle centre in the order of cognitions1 is excited1 indeed1 by itself1 and produces that which falls under cogitation3 but because it is not separate from body1 it deduces into partition1 interval1 and figure1 the objects of its knowledge1 from the indivisibility of an intellectual life4 ,hence1 whatever it knows1 is a certain impression and form of intelligence4 ,for it understands the circle1 together with its interval1 void1 indeed1 of external matter1 but possessing intelligible matter4 ,on this account1 like sensible matter1 it does not contain one circle only3 for we behold in its receptacle1 distance1 together with the more and the less1 and a multitude of circles and triangles4 ,if then an universal nature is distributed in sensible circles1 since each of these completes a circular figure1 and they are all mutually similar1 subsisting in one reason1 but differing in magnitudes or subjects3 in like manner1 there is a common something in the circles1 which subsist in the receptacle of the phantasy1 of which all its circles participate1 and according to which they all possess the same form2 but in the phantasy they possess but one difference only1 that of magnitude4 ,for when you imagine many circles about the same centre1 they all of them exist in one immaterial subject and life1 which is inseparable from a simple body1 which1 by the possession of interval1 exceeds an essence destitute of parts2 but they differ in magnitude and parvitude1 and because they are contained and contain4 ,hence1 that universal is twofold1 which is understood as subsisting in the many3 one1 indeed1 in sensible forms2 but the other in such as are imaginative4 ,and the reason of a circular and triangular figure1 and of figure universal1 is twofold4 ,the one subsisting in intelligible1 but the other in sensible matter4 ,but prior to these is the reason which resides in cogitation1 and that which is seated in nature herself4 ,the former being the author of imaginative circles1 and of the one form which they contain2 but the other1 of such as are sensible4 ,for there are circles existing in the heavens1 and universally those produced by nature1 the reason of which does not fall under a cogitative distribution4 ,for in incorporeal causes1 things possessing interval1 are distinguished by no intervals3 such as are endued with parts1 subsist without parts3 and magnitudes without the diffusion of magnitude1 as on the contrary in corporeal causes1 things without parts subsist divisibly1 and such as are void of magnitude with the extension of magnitude4 ,hence1 the circle resident in cogitation1 is one1 simple and free from interval3 and magnitude itself is there destitute of magnitude2 and figure expressed by no figure3 for such are reasons separate from matter4 ,but the circle subsisting in the phantasy1 is divisible1 figured1 endued with interval1 not one only1 but one and many1 nor form alone1 but distributed with form4 ,and the circle1 in sensible objects1 is composite1 distant with magnitude1 diminished by a certain reason1 full of ineptitude1 and very remote from the purity of immaterial natures4 ,we must therefore say1 that geometry1 when it asserts any thing of circle and diameter1 and of the passions and affections which regard the circle2 as of contacts1 divisions1 and the like3 neither teaches nor discourses concerning sensible forms1 7since it endeavours to separate us from these71 nor yet concerning the form resident in cogitation1 7for here the circle is one1 but geometry discourses of many1 proposing something of each1 and contemplating the same of all3 and here it is indivisible1 but the geometric circle is divisible72 but we must confess1 that it considers universal itself2 yet as distributed in imaginative circles4 ,and that it beholds1 indeed1 one circle#ih3 and by the medium of another1 contemplates the circle resident in the depths of cogitation3 but by another1 different from the preceding1 fabricates the fair variety of its demonstrations4 ,for since cogitation is endued with reasons1 but cannot behold them contractedly1 separated from material figure2 it distributes and removes them1 and draws them forth seated in the shadowy bosom of the phantasy1 and placed in the vestibules of primary forms2 revolving in it1 or together with it1 the knowledge of these3 loving1 indeed1 a separation from sensibles1 but finding imaginative matter proper for the reception of its universal forms4 ,hence1 its intellection does not subsist without the phantasy4 ,and the compositions and divisions of figures are imaginative2 and their knowledge is the way which leads us to that essence pursued by cogitation3 but cogitation itself1 does not yet arrive at this stable essence1 while it looks abroad to externals1 contemplates its internal forms according to these1 uses the impressions of reasons1 and is moved from itself to external and material forms4 ,but if it should ever be able to return to itself1 when it has contracted intervals and impressions1 and beholds multitude without impression1 and subsisting uniformly2 then it will excellently perceive geometrical reasons1 void of division and interval1 essential and vital1 of which there is a copious variety4 ,and this energy will be the best end of the geometric study2 and truly the employment of a ,mercurial gift1 bringing it back as from a certain ,calypso1 and her detaining charms1 to a more intellectual knowledge2 and freeing it from those forming apprehensions with which the mirror of the phantasy is replete4 ,indeed1 it is requisite that a true geometrician should be employed in this meditation1 and should establish1 as his proper end1 the excitation and transition from the phantasy to cogitation alone2 and that he should accomplish this by separating himself from intervals1 and the passive intellect to that energy which cogitation contains4 ,for by this means he will perceive all things without an interval1 the circle and diameter without a part1 the polygons in the circle1 all in all1 and yet every one separate and apart4 ,since1 on this account1 we exhibit also in the phantasy1 both circles inscribed in polygons1 and polygons in circles2 imitating the alternate exhibition of reasons destitute of parts4 ,hence1 therefore1 we describe the constitutions1 the origin1 divisions1 positions1 and applications of figures3 because we use the phantasy1 and distances of this kind proceeding from its material nature2 since form itself is immoveable1 without generation1 indivisible1 and free from every subject4 ,but whatever form contains occultly1 and in an indistant manner1 is produced into the phantasy subsisting with intervals1 divisibly and expanded4 ,and that which1 indeed1 produces the forms of geometric speculation1 is cogitation3 but that from which they are produced1 is the form resident in cogitation3 and that in which the produced figure resides is what is called the passive intellect4 ,which folds itself about the impartibility of true intellect1 separates from itself the power of pure intelligence free from interval2 conforms itself according to all formless species1 and becomes perfectly every thing from which cogitation itself1 and our indivisible reason consists4 ,and thus much concerning the geometric matter1 as we are not ignorant of whatever ,porphyry the ,philosopher has observed in his miscellanies1 and whatever many of the ,platonists describe4 ,but we think that the present discussions are more agreeable to geometric dissertations1 and to ,plato himself1 who subjects to geometry the objects of cogitation4 ,for these mutually agree among themselves2 because the causes1 indeed1 of geometrical forms1 by which cogitation produces demonstrations1 pre-exist in demonstration itself3 but the particular figures which are divided and compounded1 are situated in the receptacle of the phantasy4 ,,chap4 ,,ii4 ,what kind of ,science ,geometry is4 ,but let us now speak of that science which possesses a power of contemplating the universal forms participated by imaginative matter4 ,geometry1 therefore1 is endued with the knowledge of magnitudes and figures1 and of the terms and reasons subsisting in these2 together with the passions1 various positions and motions which are contingent about these4 ,for it proceeds1 indeed1 from an impartible point1 but descends even to solids1 and finds out their multiform diversities4 ,and again1 runs back from things more composite1 to things more simple1 and to the principles of these3 since it uses compositions and resolutions1 always beginning from suppositions1 and assuming its principles from a previous science2 but employing all the dialectic ways4 ,in principles1 by the divisions of forms from their genera1 and by defining its orations4 ,but in things posterior to principles1 by demonstrations and resolutions4 ,as likewise1 it exhibits things more various1 proceeding from such as are more simple1 and returning to them again4 ,besides this1 it separately discourses of its subjects2 separately of its axioms2 from which it rises to demonstrations2 and separately of essential accidents1 which it shews likewise are resident in its subjects4 ,for every science has1 indeed1 a genus1 about which it is conversant1 and whose passions it proposes to consider3 and besides this1 principles1 which it uses in demonstrations2 and essential accidents4 ,axioms1 indeed1 are common to all sciences 7though each employs them in its peculiar subject matter71 but genus and essential accident vary according to the sciential variety4 ,the subjects of geometry are therefore1 indeed1 triangles1 quadrangles1 circles1 and universally figures and magnitudes1 and the boundaries of these4 ,but its essential accidents are divisions1 ratios1 contacts1 equalities1 applications1 excesses1 defects1 and the like4 ,but its petitions and axioms1 by which it demonstrates every particular are1 this1 to draw a right line from any point to any point2 and that1 if from equals you take away equals1 the remainders will be equal2 together with the petitions and axioms consequent to these4 ,hence1 not every problem nor thing sought is geometrical1 but such only as flow from geometric principles4 ,and he who is reproved and convicted from these1 is convinced as a geometrician4 ,but whoever is convinced from principles different from these1 is not a geometrician1 but is foreign from the geometric contemplation4 ,but the objects of the non-geometric investigation1 are of two kinds4 ,for the thing sought for1 is either from entirely different principles1 as we say that a musical enquiry is foreign from geometry1 because it emanates from other suppositions1 and not from the principles of geometry3 or it is such as uses1 indeed1 geometrical principles1 but at the same time perversely1 as if any one should say1 that parallels coincide4 ,and on this account1 geometry also exhibits to us instruments of judging1 by which we may know what things are consequent to its principles1 and what those are which fall from the truth of its principles3 for some things attend geometrical1 but others arithmetical principles4 ,and why should we speak of others1 since they are far distant from these8 ,for one science is more certain than another 7as ,aristotle says#ii7 that1 indeed1 which emanates from more simple suppositions1 than that which uses more various principles2 and that which tells the why1 than that which knows only the simple existence of a thing2 and that which is conversant about intelligibles1 than that which touches and is employed about sensibles4 ,and according to these definitions of certainty1 arithmetic is1 indeed1 more certain than geometry1 since its principles excel by their simplicity4 ,for unity is void of position1 with which a point is endued4 ,and a point1 indeed1 when it receives position1 is the principle of geometry3 but unity1 of arithmetic4 ,but geometry is more certain than spherics2 and arithmetic1 than music4 ,for these render universally the causes of those theorems1 which are contained under them4 ,again1 geometry is more certain than mechanics1 optics1 and catoptrics4 ,because these discourse only on sensible objects4 ,the principles1 therefore1 of geometry and arithmetic1 differ1 indeed1 from the principles of other sciences2 but the hypotheses of these two1 alternately differ and agree according to the difference we have already described4 ,hence1 also1 with respect to the theorems which are demonstrated in these sciences1 some are1 indeed1 common to them1 but others peculiar4 ,for the theorem which says1 every proportion may be expressed1 alone belongs to arithmetic2 but by no means to geometry3 since this last science contains things which cannot be expressed#ajj4 ,that theorem also1 which affirms1 that the gnomons of quadrangles are terminated according to the least#aja1 is the property of arithmetic3 for in geometry1 a minimum cannot be given4 ,but those things are peculiar to geometry1 which are conversant about positions2 for numbers have no position3 which respect contacts2 for contact is found in continued quantities3 and which are conversant about ineffable proportions2 for where division proceeds to infinity1 there also that which is ineffable is found#ajb4 ,but things common to both these sciences1 are such as respect divisions1 which ,euclid treats of in the second book2 except that proposition which divides a right line into extreme and mean proportion#ajc4 ,again1 of these common theorems1 some1 indeed1 are transferred from geometry into arithmetic2 but others1 on the contrary1 from arithmetic into geometry3 and others similarly accord with both1 which are derived into them from the whole mathematical science4 ,for the permutation1 indeed1 conversions1 compositions1 and divisions of ratios are1 after this manner1 common to both4 ,but such things as are commensurable1 arithmetic first beholds2 but afterwards geometry1 imitating arithmetic4 ,from whence1 also1 it determines such things to be commensurables of this kind1 which have the same mutual ratio to one another1 as number to number2 because commensurability principally subsists in numbers4 ,for where number is1 there also that which is commensurable is found2 and where commensurable is1 there also number4 ,lastly1 geometry first inspects triangles and quadrangles3 but1 arithmetic1 receiving these from geometry1 considers them according to proportion4 ,for in numbers1 figures reside in a causal manner4 ,being excited1 therefore1 from effects1 we pass to their causes1 which are contained in numbers4 ,and at one time1 we indifferently behold the same accidents1 as when every polygon is resolved by us into triangles#ajd3 but1 at another time1 we are content with what is nearest to the truth1 as when we find in geometry one quadrangle the double of another1 but not finding this in numbers1 we say that one square is double of another1 except by a deficience of unity4 ,as for instance1 the square from #g1 is double the square from #e1 wanting one4 ,but we have produced our discussion to this length1 for the purpose of evincing the communion and difference in the principles of these two sciences4 ,since it belongs to a geometrician to survey from what common principles common theorems are divided2 and from what principles such as are peculiar proceed2 and thus to distinguish between the geometrical1 and non-geometrical1 referring each of them to different sciences4 ,,chap4 ,,iii4 ,from whence the whole of ,geometry originated1 how far it proceeds1 and in what its ,utility consists4 ,but1 beginning still higher1 let us contemplate the whole of geometry1 from whence it originated1 and how far it proceeds in its energies3 for thus we shall properly perceive the ornament which it contains4 ,indeed1 it is necessary to understand that it is extended through the universality of things3 that it accommodates its animadversions#aje to all beings2 and contains in itself the forms of all things3 that1 according to its supreme part1 and which is endued with the highest power of intelligence1 it surveys true beings2 and teaches by images the properties of divine ornaments1 and the powers of intellectual forms3 for it contains the reasons of these also in its peculiar contemplations4 ,and it exhibits what figures are convenient to the god1 to primary essences1 and to the natures of souls4 ,but1 according to its middle cognitions1 it evolves cogitative reasons2 explains and beholds the variety which they contain2 exhibits their existence1 and inherent passions2 as also1 their communities and diversities4 ,from which1 indeed1 it comprehends1 in terminated bounds1 the imaginative formations of figures1 and reduces them to the essential substance of reasons4 ,but1 according to the third propagations of cogitative intelligence1 it considers nature1 and delivers the manner in which the forms of sensible elements1 and the powers which they contain1 are previously received according to cause1 in the reasons themselves4 ,for it possesses1 indeed1 the images of universal intelligible genera2 but the exemplars of such as are sensible3 and completes its own essence1 according to such things as are subject to cogitation4 ,and through these1 as through proper mediums1 it ascends and descends to those universals which truly are1 and to sensible forms which are in a state of perpetual formation4 ,but always geometrically philosophising concerning the things which are1 it comprehends in all the proportions of virtues1 the images of intellectual1 animal1 and natural concerns4 ,and it delivers1 in an orderly manner1 all the ornaments of republics3 and exhibits in itself their various mutations4 ,such then are its energies arising from a certain immaterial power of cognition3 but when it touches upon matter1 it produces from itself a multitude of sciences2 such as geodsia1 mechanics1 and perspective3 by which it procures the greatest benefit to the life of mortals4 ,for it constructs by these sciences1 war-instruments1 and the bulwarks of cities2 and makes known the circuits of mountains1 and the situations of places4 ,lastly1 it instructs us in measures3 at one time of the diversified ways of the earth2 and at another1 of the restless paths of the deep4 ,add too1 that it constructs balances and scales1 by which it renders to cities a sure equality according to the invariable standard of number4 ,likewise1 it clearly expresses1 by images1 the order of the whole orb of the earth2 and by these1 manifests many things incredible to mankind1 and renders them credible to all4 ,such1 indeed1 as ,hiero of ,syracuse is reported to have said of ,archimedes#ajf1 when he had fabricated a ship furnished with three sails1 which he had prepared to send to ,ptolemy king of ,egypt4 ,for when all the ,syracusians together1 were unable to draw this ship1 ,archimedes enabled ,hiero to draw it himself1 without any assistance from others4 ,but he1 being astonished1 said1 ,from this day1 ,archimedes shall be believed in whatever he shall affirm4 ,they also report1 that ,gelo said the same1 when ,archimedes discovered the weight of the several materials from which his crown was composed1 without dissolving their union4 ,and such are the narrations which many of the ancients have delivered to our memory1 who were willing to speak in praise of the mathematics3 and1 on this account1 we have placed before the reader1 for the present1 a few out of the many1 as not foreign from our design of exhibiting the knowledge and utility of geometry4 ,,chap4 ,,iv4 ,on the ,origin of ,geometry1 and its ,inventors4 ,but let us now explain the origin of geometry1 as existing in the present age of the world4 ,for the demoniacal ,aristotle#ajg observes1 that the same opinions often subsist among men1 according to certain orderly revolutions of the world3 and that sciences did not receive their first constitution in our times1 nor in those periods which are known to us from historical tradition1 but have appeared and vanished again in other revolutions of the universe2 nor is it possible to say how often this has happened in past ages1 and will again take place in the future circulations of time4 ,but1 because the origin of arts and sciences is to be considered according to the present revolution of the universe1 we must affirm1 in conformity with the most general tradition1 that geometry was first invented by the ,egyptians1 deriving its origin from the mensuration of their fields3 since this1 indeed1 was necessary to them1 on account of the inundation of the ,nile washing away the boundaries of land belonging to each4 ,nor ought it to seem wonderful that the invention of this as well as of other sciences1 should receive its commencement from convenience and opportunity4 ,since whatever is carried in the circle of generation1 proceeds from the imperfect to the perfect4 ,a transition1 therefore1 is not undeservedly made from sense to consideration1 and from this to the nobler energies of intellect#ajh4 ,hence1 as the certain knowledge of numbers received its origin among the ,phnicians1 on account of merchandise and commerce1 so geometry was found out among the ,egyptians from the distribution of land4 ,when ,thales1 therefore1 first went into ,egypt1 he transferred this knowledge from thence into ,greece3 and he invented many things himself1 and communicated to his successors the principles of many4 ,some of which were1 indeed1 more universal1 but others extended to sensibles4 ,after him ,ameristus1 the brother of ,stesichorus the poet1 is celebrated as one who touched upon1 and tasted the study of geometry1 and who is mentioned by ,hippias the ,elean1 as restoring the glory of geometry4 ,but after these1 ,pythagoras changed that philosophy1 which is conversant about geometry itself1 into the form of a liberal doctrine1 considering its principles in a more exalted manner2 and investigating its theorems immaterially and intellectually2 who likewise invented a treatise of such things as cannot be explained#aji in geometry1 and discovered the constitution of the mundane figures4 ,after him1 ,anaxagoras the ,clazomenian succeeded1 who undertook many things pertaining to geometry4 ,and ,oenopides the ,chian1 was somewhat junior to ,anaxagoras1 and whom ,plato mentions in his ,rivals1 as one who obtained mathematical glory4 ,to these1 succeeded ,hippocrates1 the ,chian1 who invented the quadrature of the lunula#aaj1 and ,theodorus the ,cyrenean1 both of them eminent in geometrical knowledge4 ,for the first of these1 ,hippocrates composed geometrical elements3 but ,plato1 who was posterior to these1 caused as well geometry itself1 as the other mathematical disciplines1 to receive a remarkable addition1 on account of the great study he bestowed in their investigation4 ,this he himself manifests1 and his books1 replete with mathematical discourses1 evince3 to which we may add1 that he every where excites whatever in them is wonderful1 and extends to philosophy4 ,but in his time also lived ,leodamas the ,thasian1 ,archytas the ,tarantine1 and ,thetetus the ,athenian2 by whom theorems were increased1 and advanced to a more skilful constitution4 ,but ,neoclides was junior to ,leodamas1 and his disciple was ,leon2 who added many things to those thought of by former geometricians4 ,so that ,leon also constructed elements more accurate1 both on account of their multitude1 and on account of the use which they exhibit3 and besides this1 he discovered a method of determining when a problem1 whose investigation is sought for1 is possible1 and when it is impossible4 ,but ,eudoxus the ,cnidian1 who was somewhat junior to ,leon1 and the companion of ,plato1 first of all rendered the multitude of those theorems which are called universals more abundant2 and to three proportions added three others2 and things relative to a section1 which received their commencement from ,plato1 he diffused into a richer multitude1 employing also resolutions in the prosecution of these4 ,again1 ,amyclas the ,heracleotean1 one of ,plato's familiars1 and ,menchmus1 the disciple1 indeed1 of ,eudoxus1 but conversant with ,plato1 and his brother ,dinostratus1 rendered the whole of geometry as yet more perfect4 ,but ,theudius1 the ,magnian1 appears to have excelled1 as well in mathematical disciplines1 as in the rest of philosophy4 ,for he constructed elements egregiously1 and rendered many particulars more universal4 ,besides1 ,cyzicinus the ,athenian1 flourished at the same period1 and became illustrious in other mathematical disciplines1 but especially in geometry4 ,these1 therefore1 resorted by turns to the ,academy1 and employed themselves in proposing common questions4 ,but ,hermotimus1 the ,colophonian1 rendered more abundant what was formerly published by ,eudoxus and ,thetetus1 and invented a multitude of elements1 and wrote concerning some geometrical places4 ,but ,philippus the ,mendan#aaa1 a disciple of ,plato1 and by him inflamed in the mathematical disciplines1 both composed questions1 according to the institutions of ,plato1 and proposed as the object of his enquiry whatever he thought conduced to the ,platonic philosophy4 ,and thus far historians produce the perfection of this science4 ,but ,euclid was not much junior to these1 who collected elements1 and constructed many of those things which were invented by ,eudoxus2 and perfected many which were discovered by ,thetetus4 ,besides1 he reduced to invincible demonstrations1 such things as were exhibited by others with a weaker arm4 ,but he lived in the times of the first ,ptolemy3 for ,archimedes mentions ,euclid1 in his first book1 and also in others4 ,besides1 they relate that ,euclid was asked by ,ptolemy1 whether there was any shorter way to the attainment of geometry than by his elementary institution1 and that he answered1 there was no other royal path which led to geometry4 ,euclid1 therefore1 was junior to the familiars of ,plato1 but more ancient than ,eratosthenes and ,archimedes 7for these lived at one and the same time1 according to the tradition of ,eratosthenes7 but he was of the ,platonic sect1 and familiar with its philosophy3 and from hence he appointed the constitution of those figures which are called ,platonic#aab1 as the end of his elementary institutions4 ,,chap4 ,v4 ,what ,mathematical ,volumes ,euclid composed4 ,there are1 therefore1 many other mathematical volumes of this man1 full of admirable diligence1 and skilful consideration3 for such are his ,optics#aac1 and ,catoptrics3 and such also1 are his elementary institutions1 which conduce to the attainment of music#aad2 and his book concerning divisions#aae4 ,but his geometrical institution of the ,elements is especially admirable1 on account of the order and election of those theorems and problems1 which are distributed through the ,elements4 ,for he does not assume all which might be said1 but that only which could be delivered in an elementary order4 ,besides this1 he exhibits modes of syllogisms of every kind2 some1 indeed1 receiving credibility from causes1 but others proceeding from certain signs2 but all of them invincible and sure1 and accommodated to science4 ,but1 besides these1 he employs all the dialectic ways1 dividing1 indeed1 in the inventions of forms2 but defining in essential reasons3 and again1 demonstrating in the progressions from principles to things sought1 but resolving in the reversions from things sought to principles4 ,besides this1 we may view in his geometrical elements1 the various species of conversions1 as well of such as are simple as of such as are more composite4 ,and again1 what wholes may be converted with wholes3 what wholes with parts2 and on the other hand1 what as parts with parts#aaf4 ,besides this1 we must say1 that in the continuation of inventions1 the dispositions and order of things preceding and following1 and in the power with which he treats every particular1 he is not deceived1 as if falling from science1 and carried to its contrary1 falsehood and ignorance4 ,but because we may imagine many things as adhering to truth1 and which are consequent to principles producing science1 which nevertheless tend to that error which flows from the principles1 and which deceives ruder minds1 he has also delivered methods of the perspicacious prudence belonging to these4 ,from the possession of which1 we may exercise those in the invention of fallacies1 who undertake this inspection1 and may preserve ourselves from all deception4 ,and this book1 by which he procures us this preparation1 is inscribed 1 or1 concerning fallacies#aag4 ,because he enumerates in order their various modes1 and in each exercises our cogitation with various theorems4 ,and he compares truth with falsehood1 and adapts the confutation of deception to experience itself4 ,this book1 therefore1 contains a purgative and exercising power4 ,but the institution of his elementary1 skilful contemplation of geometrical concerns1 possesses an invincible and perfect narration4 ,,chap4 ,,vi4 ,concerning the ,purport of ,geometry4 ,but1 perhaps1 some one may enquire in what the design of this treatise consists8 ,to this ,i answer1 that its design is to be distinguished as well according to the objects of enquiry1 as according to the learner4 ,and1 indeed1 regarding the subject1 we must affirm1 that all the discourse of geometry is concerning the mundane figures4 ,because it begins from such things as are simple1 but ends in the variety of their constitution4 ,and1 indeed1 it constitutes each of them separately1 but at the same time delivers their inscriptions in a sphere1 and the proportions which they contain4 ,on which account some have thought1 that the design of each of the books is to be referred to the world2 and they have delivered to our memory1 the utility which they afford us in the contemplation of the universe4 ,but distinguishing the design with respect to the learner1 we must affirm1 that its purpose is the institution of elements2 and the perfection of the learners cogitative powers in universal geometry4 ,for beginning from these1 we are enabled to understand the other parts of this science1 and to comprehend the variety which they contain4 ,and1 indeed1 without these1 the discipline of the rest1 is to us impossible and incomprehensible4 ,for such theorems as are most principal and simple1 and are most allied to first suppositions1 are here collected in a becoming order4 ,and the demonstrations of other mathematicians1 use these as most known1 and advance from these in their most complicated progressions4 ,for thus ,archimedes1 in what he has writ concerning the sphere and cylinder1 and ,apollonius1 and the rest of mathematicians1 use1 as evident principles1 the things exhibited in this treatise4 ,its purpose1 therefore1 is the institution of learners in the whole geometric science1 and to deliver the determinate constitutions of the mundane figures4 ,,chap4 ,,vii4 ,from whence the ,name of ,elementary ,institution originated1 and why ,euclid is called the ,institutor of ,elements4 ,but what gave rise to the name of elementary institution1 and of element itself1 from which elementary institution was derived8 ,to this we shall reply1 by observing1 that of theorems some are usually called elements1 but others elementary1 and others again are determined beyond the power of these4 ,hence1 an element is that whose consideration passes to the science of other things1 and from which we derive a solution of the doubts incident to the particular science we investigate4 ,for as there are certain first principles of speech1 most simple and indivisible1 which we denominate elements1 and from which all discourse is composed2 so there are certain principal theorems of the whole of geometry1 denominated elements1 which have the respect of principles to the following theorems2 which regard all the subsequent propositions1 and afford the demonstrations of many accidents essential to the subjects of geometric speculation4 ,but things elementary are such as extend themselves to a multitude of propositions1 and possess a certain simplicity and sweetness1 yet are not of the same dignity with elements2 because their contemplation is not common to all the science to which they belong1 as is the case in the following theorem1 that in triangles1 perpendiculars1 drawn from their angles to their sides1 coincide in one point#aah4 ,lastly1 whatever neither possesses a knowledge extended into multitude1 nor exhibits any thing skilful and elegant1 falls beyond the elementary power4 ,again1 an element1 as ,menchmus says1 may have a twofold definition4 ,for that which confirms1 is an element of that which is confirmed2 as the first proposition of ,euclid with respect to the second1 and the fourth with regard to the fifth4 ,and thus1 indeed1 many things may be mutually called elements one of another2 for they are mutually confirmed4 ,thus1 because the external angles of right-lined figures1 are equal to four right angles1 the multitude of internal ones equal to right angles2 and1 on the contrary1 that from this is exhibited#aai4 ,besides1 an element is otherwise called that into which1 because it is more simple1 a composite is dissolved4 ,but it must be observed1 that every element cannot be called the element of every thing3 but such as are more principal are the elements of such as are constituted in the reason of the thing effected2 as petitions are the elements of theorems4 ,and1 according to this signification of an element1 ,euclid's elements are constructed4 ,some1 indeed1 of that geometry which is conversant about planes2 but others of stereometry4 ,in the same manner1 likewise1 in arithmetic and astronomy1 many have composed elementary institutions4 ,but it is difficult1 in each science1 to chuse and conveniently ordain elements1 from which all the peculiarities of that science originate1 and into which they may be resolved4 ,and among those who have undertaken this employment1 some have been able to collect more1 but others fewer elements4 ,and some1 indeed1 have used shorter demonstrations2 but others have extended their treatise to an infinite length4 ,and some have omitted the method by an impossibility2 but others that by proportion2 and others1 again1 have attempted preparations against arguments destroying principles4 ,so that many methods of elementary institution have been invented by particular writers on this subject4 ,but it is requisite that this treatise should entirely remove every thing superfluous1 because it is an impediment to science4 ,but every thing should be chosen1 which contains and concludes the thing proposed2 for this is most convenient and useful in science4 ,the greatest care1 likewise1 should be paid to clearness and brevity2 for the contraries to these1 disturb our cogitation4 ,lastly1 it should vindicate to itself1 the universal comprehension of theorems1 in their proper bounds3 for such things as divide learning into particular fragments1 produce an incomprehensible knowledge4 ,but in all these modes1 any one may easily find1 that the elementary institution of ,euclid excels the institutions of others4 ,for its utility1 indeed1 especially confers to the contemplation of primary figures3 but the transition from things more simple to such as are more various1 and also that perception1 which from axioms possesses the beginning of knowledge1 produces clearness1 and an orderly tradition3 and the migration from first and principal theorems to the objects of enquiry1 effects the universality of demonstration4 ,for whatever he seems to omit1 may either be known by the same ways1 as the construction of a scalene and isosceles triangle#abj3 or because they are difficult1 and capable of infinite variety1 they are far remote from the election of elements1 such as the doctrine of perturbate proportions1 which ,apollonius has copiously handled3 or1 lastly1 because they may be easily constructed from the things delivered1 as from causes1 such as many species of angles and lines4 ,for these1 indeed1 were omitted by ,euclid1 and are largely discoursed of by others1 and are known from simple propositions4 ,and thus much concerning the universal elementary institution of geometry4 ,,chap4 ,,viii4 ,concerning the ,order of ,geometrical ,discourses4 ,but let us now explain the universal order of the discourses contained in geometry4 ,because then1 we assert that this science consists from hypothesis#aba1 and demonstrates its consequent propositions from definite principles 7for one science only1 ,i mean the first philosophy1 is without supposition1 but all the rest assume their principles from this7 it is necessary that he who constructs the geometrical institution of elements1 should separately deliver the principles of the science1 and separately the conclusions which flow from those principles2 and that he should render no reason concerning the nature or truth of the principles1 but should confirm by reasons1 the things consequent to these geometric principles4 ,for no science demonstrates its own principles1 nor discourses concerning them2 but procures to itself a belief of their reality1 and they become more evident to the particular science to which they belong than the things derived from them as their source4 ,and these1 indeed1 science knows by themselves2 but their consequents1 through the medium of these4 ,for thus1 also1 the natural philosopher propagates his reasons from a definite principle1 supposing the existence of motion4 ,thus too1 the physician1 and he who is skilled in any of the other sciences and arts4 ,for if any one mingles principles1 and things flowing from principles into one and the same1 he disturbs the whole order of knowledge1 and conglutinates things which can never mutually agree2 since a principle1 and its emanating consequent1 are naturally distinct from each other4 ,in the first place1 therefore 7as ,i have said71 principles in the geometric institution are to be distinguished from their consequents1 which is performed by ,euclid in each of his books2 who1 before every treatise1 exhibits the common principles of this science2 and afterwards divides these common principles into hypotheses1 petitions1 and axioms4 ,for all these mutually differ2 nor is an axiom1 petition1 and hypothesis the same1 according to the demoniacal ,aristotle2 but when that which is assumed in the order of a principle1 is indeed known to the learner1 and credible by itself1 it is an axiom3 such as1 that things equal to the same1 are mutually equal to each other4 ,but when any one1 hearing another speak concerning that of which he has no self-evident knowledge1 gives this assent to its assumption1 this is hypothesis4 ,for that a circle is a figure of such a particular kind1 we presume 7not according to any common conception7 without any preceding doctrine4 ,but when1 again1 that which is asserted was neither known1 nor admitted by the learner1 yet is assumed1 then 7says he7 we call it petition2 as the assumption that all right angles are equal4 ,but the truth of this is evinced by those who study to treat of some petition1 as of that which cannot by itself be admitted by any one4 ,and thus1 according to the doctrine of ,aristotle#abb1 are axiom1 petition1 and supposition distinguished4 ,but oftentimes1 some denominate all these hypotheses1 in the same manner as the ,stoics call every simple enunciation an axiom4 ,so that1 according to their opinion1 hypotheses also will be axioms2 but1 according to the opinion of others1 axioms will be called suppositions4 ,again1 such things as flow from principles are divided into problems and theorems4 ,the first1 indeed1 containing the origin1 sections1 ablations1 or additions of figures1 and all the affections with which they are conversant2 but the other exhibiting the accidents essential to each figure4 ,for1 as things effective of science1 participate of contemplation1 in the same manner things contemplative previously assume problems in the place of operations4 ,but formerly some of the ancient mathematicians thought that all geometrical propositions should be called theorems1 as the followers of ,speusippus and ,amphinomus1 believing1 that to contemplative sciences1 the appellation of theorems is more proper than that of problems2 especially since they discourse concerning eternal and immutable objects4 ,for origin does not subsist among things eternal3 on which account1 problems cannot have any place in these sciences2 since they enunciate origin1 and the production of that which formerly had no existence1 as the construction of an equilateral triangle1 or the description of a square on a given right line1 or the position of a right line at a given point4 ,it is better1 therefore 7say they71 to assert that all propositions are of the speculative kind2 but that we perceive their origin1 not by production1 but by knowledge1 receiving things eternal as if they were generated2 and on this account we ought to conceive all those theorematically1 but not problematically4 ,but others1 on the contrary1 think that all should be called problems2 as those mathematicians who have followed ,menchmus4 ,but that the office of problems is twofold1 sometimes1 indeed1 to procure the thing sought2 but at other times when they have received the determinate object of enquiry1 to see1 either what it is1 or of what kind it is1 or what affection it possesses1 or what its relation is to another4 ,and1 indeed1 the assertions of each are right2 for the followers of ,speusippus well perceive4 ,since the problems of geometry are not of the same kind1 with such as are mechanical4 ,for these are sensibles1 and are endued with origin1 and mutation of every kind4 ,and1 on the other hand1 those who follow ,menchmus do not dissent from truth3 since the inventions of theorems cannot by any means take place without an approach into matter2 ,i mean intelligible matter4 ,reasons1 therefore1 proceeding into this1 and giving form to its formless nature1 are not undeservedly said to be assimilated to generations4 ,for we say that the motion of our cogitation1 and the production of its inherent reasons1 is the origin of the figures situated in the phantasy1 and of the affections with which they are conversant3 for there constructions and sections1 positions and applications1 additions and ablations1 exist3 but every thing resident in cogitation1 subsists without origin and mutation4 ,there are1 therefore1 both geometrical problems and theorems4 ,but1 because contemplation abounds in geometry1 as production in mechanics1 all problems participate of contemplation2 but every thing contemplative is not problematical4 ,for demonstrations are entirely the work of contemplation2 but every thing in geometry posterior to the principles1 is assumed by demonstration4 ,hence1 a theorem is more common3 but all theorems do not require problems2 for there are some which possess from themselves the demonstration of the thing sought4 ,but others1 distinguishing a theorem from a problem1 say1 that indeed every problem receives whatever is predicated of its matter1 together with its own opposite3 but that every theorem receives1 indeed1 its symptom predicate1 but not its opposite4 ,but ,i call the matter of these1 that genus which is the subject of enquiry2 as for instance1 a triangle1 quadrangle1 or a circle3 but the symptom predicate1 that which is denominated an essential accident1 as equality1 or section1 or position1 or some other affection of this kind4 ,when1 therefore1 any one proposes to inscribe an equilateral triangle in a circle1 he proposes a problem3 for it is possible to inscribe one that is not equilateral4 ,but when any one asserts that the angles at the base of an isosceles triangle are equal1 we must affirm that he proposes a theorem2 for it is not possible that the angles at the base of an isosceles triangle should be unequal to each other4 ,on which account1 if any one forming problematically1 should say that he wishes to inscribe a right angle in a semi-circle1 he must be considered as ignorant of geometry2 since every angle in a semi-circle is necessarily a right one4 ,hence1 propositions which have an universal symptom1 attending the whole matter1 must be called theorems2 but those in which the symptom is not universal1 and does not attend its subject1 must be considered as problems4 ,as to bisect a given terminated right line1 or to cut it into equal parts3 for it is possible to cut it into unequal parts4 ,to bisect every rectilinear angle1 or divide it into equal parts2 for a division may be given into unequal parts4 ,on a given right line to describe a quadrangle2 for a figure that is not quadrangular may be described4 ,and1 in short1 all of this kind belong to the problematical order4 ,but the followers of ,zenodotus1 who was familiar with the doctrine of ,oenopides1 but the disciple of ,andron1 distinguish a theorem from a problem1 so far as a theorem enquires what the symptom is which is predicated of the matter it contains2 but a problem enquires what that is1 the existence of which is granted4 ,from whence the followers of ,possidonius define a theorem a proposition1 by which it is enquired whether a thing exists or not2 but a problem1 a proposition1 in which it is enquired what a thing is1 or the manner of its existence4 ,and they say that we ought to form the contemplating proposition by enunciating1 as that every triangle has two sides greater than the remaining one1 and that the angles at the base of every isosceles triangle are equal3 but we must form the problematical proposition1 as if enquiring whether a triangle is to be constructed upon this right line4 ,for there is a difference1 say they1 absolutely and indefinitely1 to enquire whether the thing proposed is from a given point to erect a right line at right angles to a given line1 and to behold what the perpendicular is4 ,and thus1 from what has been said1 it is manifest there is some difference between a problem and a theorem4 ,but that the elementary institution of ,euclid1 also1 consists partly of problems1 and partly of theorems1 will be manifest from considering the several propositions4 ,since1 in the conclusion of his demonstrations1 he sometimes adds 7which was to be shewn7 sometimes 7which was to be done7 the latter sentence being the mark or symbol of problems1 and the former of theorems4 ,for although1 as we have said1 demonstration takes place in problems1 yet it is often for the sake of generation2 for we assume demonstration in order to shew1 that what was commanded is accomplished3 but sometimes it is worthy by itself1 since the nature of the thing sought after may be brought into the midst4 ,but you will find ,euclid sometimes combining theorems with problems1 and using them alternately1 as in the first book2 but sometimes abounding with the one and not the other4 ,for the fourth book is wholly problematical2 but the fifth is entirely composed from theorems4 ,and thus much concerning the order of geometrical propositions4 ,,chap4 ,,ix4 ,concerning the ,design of the first ,book1--its ,division1--and a previous ,admonition to the ,reader4 ,but1 after these considerations1 when we have determined the design of the first book1 and have exhibited its division1 we shall enter upon the treatise of the definitions4 ,the design1 then1 of this book1 is to deliver the principles of the contemplation of right lines4 ,for though a circle1 and its consideration1 is more excellent than the essence and knowledge of right lines1 yet the doctrine concerning these is more adapted to us1 who are hastening to transfer our cogitation from more imperfect and sensible natures1 to such as are intelligible4 ,for1 indeed1 right lined figures are proper to sensibles1 but a circle to intelligibles4 ,because that which is simple1 uniform1 and definite1 is proper to the nature of the things which are3 but that which is various1 and which increases indefinitely from the number of its containing sides1 regards the fluctuating essence of sensible particulars4 ,hence1 in this book1 the first and most principal of right lined figures are delivered2 ,i mean the triangle and parallelogram4 ,for in these1 as under their proper genus1 the causes of the elements are contained3 viz4 the isosceles and scalene1 and those which are formed from these1 the equilateral triangle1 and the quadrangle1 from which the four figures of the elements are composed4 ,we shall find1 therefore1 as well the origin of the equilateral triangle as of the quadrangle2 of the last1 indeed1 upon1 but of the first from a given right line4 #abc;,an equilateral triangle1 therefore1 is the proximate cause of the three elements1 fire1 air1 and water3 but a quadrangle is annexed to earth4 ,and lastly1 the design of the first book is adapted to the whole treatise1 and confers to the universal knowledge of the mundane elements4 ,besides1 it instructs learners in the science concerning right-lined figures2 since it rightly invents1 and accurately collects1 the first principles of these4 ,but this book is divided into three greatest parts1 of which the first declares the origin and properties of triangles1 as well according to angles1 as also according to sides4 ,besides1 it makes mutual comparisons of these1 and beholds every one by itself4 ,for receiving one triangle1 sometimes it considers the angles from the sides2 but sometimes the sides from the angles3 and this according to equality and inequality4 ,and supposing two triangles1 it discovers the same property again1 by various methods4 ,but the second part combines the contemplation of parallelograms1 describing their properties and generations4 ,and the third part shews the communication of triangles and parallelograms1 both in symptoms and mutual comparisons4 ,for it shews that triangles and parallelograms constituted on the same and on equal bases1 are affected with the same passions2 and by complication1 when both stand upon one base3 and again1 after what manner a parallelogram may be made equal to a triangle2 and lastly1 concerning the proportion which in right angled triangles1 the square made from the side subtending1 has to the squares containing the right angle4 ,and such is the division of the first ,book4 ,but1 previous to our enquiry into each of these parts1 we think it requisite to admonish the reader1 that he must not require of us1 those small assumptions1 and cases1 and whatever else there may be of that kind1 which has been divulged by our predecessors4 ,for we are satiated with these1 and shall1 therefore1 but rarely adopt them in our discourse4 ,but whatever has a more difficult contemplation1 and regards universal philosophy1 of this we shall make a particular relation3 imitating the ,pythagoreans1 with whom this nigma was common1 8a#abd figure and a step3 but not a figure and three oboli10 shewing by this1 that it is requisite to pursue that philosophy which ascends every theorem by a step1 and raises the soul on high2 but does not suffer it to remain among sensibles1 to fill up the use attendant on mortals1 and1 consulting for this1 to neglect the elevation which rises from hence to an intelligible essence4 ,,definitions4 ,,definition ,i4 ,a ,,point is that which has no ,,parts4 ,that geometry1 according to the transition which takes place from things more composite to such as are more simple1 runs from body1 which is diffused into distance by three dimensions1 to a superficies by which it is bounded2 but from superficies to a line1 the boundary of superficies2 and from a line to a point destitute of all dimension1 has been often said1 and is perfectly manifest4 ,but because these terms1 in many places1 on account of their simplicity1 appear to be more excellent than the nature of composites2 but in many1 as when they subsist in things which they terminate1 they are similar to accidents1 it is necessary to determine in what genera of beings each of these may be beheld#abe4 ,i say then1 that such things as are destitute of matter1 which subsist in separate reasons1 and in those forms which are placed under themselves1 are always allotted a subsistence of more simple essences1 superior to the subsistence of such as are more composite4 ,on this account1 both in intellect1 and in the ornaments1 as well of the middle kind as among those peculiar to the soul1 and in natures themselves1 the terms which proximately vivify bodies1 excel according to essence the things which are terminated2 and are more impartible1 more uniform1 and more primary than these4 ,for in immaterial forms1 unity is more perfect than multitude2 that which is impartible1 than that which is endued with unbounded progression2 and that which terminates1 than that which receives bound from another4 ,but such things as are indigent of matter1 and abide in others1 and degenerate from the perfection of their essence1 which are scattered about subjects1 and have an unnatural union1 are allotted more composite reasons1 prior to such as are more simple4 ,hence1 things which appear in the phantasy invested with form1 and the matter of the figures which the phantasy contains1 and whatever in sensibles is generated by nature1 have1 in a preceding order1 the reasons of the things terminated2 but the reasons which terminate1 in a following and adventitious rank#abf4 ,for lest that which is distributed into three dimensions1 should be extended into infinite magnitude1 either according to intelligence or sense1 it was every way terminated by superficies4 ,and lest a plane superficies should conceal itself in an infinite progression1 a line approaching opposed its diffusion1 and gave bound to its indefinite extension4 ,and1 in like manner1 a point limited the progressions of a line2 composite natures deriving their subsistence from such as are simple4 ,for this also is again manifest1 that in separate forms the reasons of terms subsist in themselves1 but not in those which are terminated2 and abiding such as they are in reality1 possess a power of constituting secondary natures4 ,but1 in inseparable forms they give themselves up to things which are terminated1 reside in them1 become1 as it were1 their parts1 and are replenished with baser natures4 ,on which account1 that which is impartible is there endued with a partible essence1 and that which is void of latitude is diffused into breadth4 ,and terms are no longer able to preserve their simplicity and purity4 ,for since they abide in another1 they necessarily change their own nature into the matter of their containing subject4 ,matter1 indeed1 disturbs the perfection of these1 and causes the reason of a plane to become a profound plane2 but obscuring the one dimension of a line1 causes it to be every way partible2 and gives corporeity to the indivisibility of a point1 and separates it together with the natures which it terminates4 ,for all these reasons falling into matter1 the one kind from cogitation into intelligible matter1 but the other from nature into that which is sensible1 are replenished with their containing subjects2 and depart from their own simplicity1 into foreign compositions and intervals4 ,but here a doubt arises how all these1 existing in intellect and soul in an impartible manner1 and without any dimension1 are distributed into matter1 some indeed1 principally1 but others on account of its nature8 ,shall we say that there is a certain order in immaterial forms1 so that some are allotted the first1 some the middle1 and others the last place2 and that of forms some are more uniform1 but that others are more multiplied2 and that some have their powers collected together1 but others tending into interval2 and that some1 again1 border upon bound1 but that others are proximate to infinity8 ,for though all participate of these two principles1 yet some originate from bound1 but others from infinity1 of which they more largely participate4 ,hence1 a point is entirely impartible1 since it subsists according to bound1 yet it occultly contains an infinite power1 by which it produces every interval1 and the progression of all intervals1 unfolds its infinite power4 ,but body1 and the reason of body1 participates more of an infinite nature2 on which account it is among the number of things terminated by another1 and divisible in infinitum1 according to all dimensions4 ,but the mediums between these1 according to the distance of the extremes1 are either among the number of things which have an abundance of bound2 or among such as have an affluence of infinity3 on which account they both terminate and are terminated4 ,for1 indeed1 so far as they consist from bound1 they are able to terminate others2 but so far as they participate of infinity1 they are indigent of termination from others1 ,hence1 since a point is also a bound1 it preserves its proper power in participation3 but since it likewise contains infinity occultly1 and is compelled to be every where present with the natures which it terminates1 it resides with them infinitely4 ,and1 because among immaterial forms there was a certain infinite power capable of producing things distant from each other by intervals1 a point is present with its participants in capacity4 ,for infinity in intelligibles is the primary cause and prolific power of the universe2 but in material natures it is imperfect1 and is alone all things in dormant capacity4 ,and in short1 those forms which1 on account of their simplicity and impartibility1 hold a superior rank among principles1 preserve1 indeed1 7in conformity to their nature17 their own property in their participations1 but become worse than more composite reasons4 ,for matter is able to participate these more clearly1 and to be prepared for their reception1 rather than that of the most simple causes of beings4 ,on which account1 the vestigies of separate principles descend into matter2 but the participations of those in a second and third order1 become more conspicuous4 ,hence1 matter participates more of the cause of body1 than of a plane2 and of this more than the form of a line2 and of this still more than that of a point1 which contains all these1 and is the boundary of them all4 ,for the reason of a point presides over this whole series1 unites and contains all partible natures1 terminates their progressions1 produces them all by its infinite power1 and comprehends them in its indivisible bound4 ,on which account also1 in the images of immaterial forms1 some are the boundaries of others2 but a point is the limit of them all4 ,but that we must not think with the ,stoics1 that these boundaries of bodies alone subsist from cogitation2 but that there are certain natures of this kind among beings1 which previously contain the demiurgical reasons of things1 we shall be enabled to remember1 if we regard the whole world1 the convolutions of its parts1 the centres of those convolutions1 and the axes which penetrate through the whole of these revolving circles4 ,for the centres subsist in energy1 since they contain the spheres1 preserve them in their proper state1 unite their intervals1 and bind and establish to themselves the powers which they possess4 ,but the axes themselves being in an immoveable position1 evolve the spheres1 give them a circular motion1 and a revolution round their own abiding nature4 ,and the poles of the spheres1 which both terminate the axes1 and bind in themselves the other convolutions1 do they not perspicuously evince1 that points are endued with demiurgical and capacious powers1 that they are perfective of every thing distant by intervals1 and are the sources of union1 and an unceasing motion8 ,from whence1 indeed1 ,plato#abg also says1 that they have an adamantine subsistence2 shewing by this1 the immutable1 eternal1 and stable power of their essence1 ever preserving itself in the same uniform mode of existence4 ,he adds too1 that the whole spindle of the ,fates1 is turned about these1 and leaps round their coercive union4 ,but other more recondite and abstruse discourses affirm1 that the demiurgus presides over the world1 seated in the poles1 and1 by his divine love1 converting the universe to himself4 ,but the ,pythagoreans thought that the pole should be called the ,seal of ,rhea#abh2 because the zoogonic1 or vivific goddess1 pours through these into the universe1 an inexplicable and efficacious power4 ,and the centre they called the prison of ,jupiter2 because1 since ,jupiter has placed a demiurgical guard in the bosom of the world1 he has firmly established it in the midst4 ,for1 indeed1 the centre abiding1 the universe possesses its immoveable ornament1 and unceasing convolution3 and the gods who preside over the poles1 obtain a power collective of divisible natures1 and unific of such as are multiplied3 and those who are allotted the government of the axes1 restrain and eternally evolve their perpetual convolutions4 ,and1 if it is lawful to offer our own opinion on this subject1 the centres and poles of all the spheres are the symbols of the conciliating gods1 shadowing forth their imperceptible and unifying composition4 ,but the axes express the coherencies of the universal ornaments2 and are endued with a power of comprehending the mundane integrities and periods1 in the same manner as their presiding deities1 of such as are intellectual4 ,but the spheres themselves are images of the gods1 called perfectors of works1 copulating the principle with the end1 and excelling all figures in simplicity1 similitude1 and perfection4 ,but we have been thus prolix1 that we might evince the power of impartibles1 and of the terms which the world contains1 and that so far as they bear an image of primary and most principal causes1 they are allotted the most excellent order in the universe4 ,for centres and poles are not of the same kind with things which are terminated2 but they subsist in energy1 and possess an essence1 and perfect power1 which pervades through all partible natures4 ,but many beholding those terms which imperfectly subsist in terminated essences1 consider them as endued with a slender subsistence2 and some indeed say1 that they are alone separated from sensibles by thought2 but others1 that they have an essence no where but in our thoughts4 ,however1 since the forms of all these are found both in the nature of intellect1 in the ornaments of soul1 in the nature of things1 and in inferior bodies1 let us consider how1 according to the order they contain1 they subsist in the genera of beings4 ,and indeed1 all of them pre-exist in intellect1 but in an impartible and uniform manner3 so that they all subsist according to one form1 the reason of a point1 which exists occultly and impartibly4 ,but they all subsist in soul according to the form of a line3 on which account ,timus also composes the soul from right and circular lines3 for every circle is a line alone#abi4 ,but they all subsist in natures1 according to the reason of a plane2 and on this account1 ,plato commands us to manifest those natural reasons1 which are endued with a power of constituting bodies by a plane4 ,and the resolution of bodies into planes leads us to the proximate cause of appearances4 ,lastly1 they all subsist in bodies1 but in a corporeal manner2 since all forms have their being in these1 according to the partible nature of bodies4 ,hence1 all of them appear every where1 and each according to its proper order2 and diversity arises from pre-dominating power4 ,the point1 indeed1 is every where impartible1 and when that which is divisible into parts1 excels according to the diminution of beings1 it vindicates to itself1 an illustrious subsistence of partible natures4 ,and sometimes the point is entirely superior1 according to the excellence of cause2 but sometimes it is connected with divisibles1 and sometimes it is allotted in them an adventitious existence2 and1 as if swallowed up by the partition of the lowest natures1 loses its own proper impartibility4 ,as1 therefore1 with respect to the monad1 one#acj is the mother of number1 but the other is as matter spread under1 and the receptacle of numbers2 and each of them a principle1 7yet neither of them is number71 but in a different respect3 in the same manner a point also1 is partly the parent and author of magnitudes2 but is partly a principle in another respect1 and not according to a generative cause4 ,but is a point1 then1 the only impartible8 ,or may we affirm this of the now in time1 and of unity in numbers8 ,shall we not say1 that to the philosopher1 indeed1 discoursing concerning the universality of things1 it is proper to behold every thing1 however falling under distribution2 but that to him who is endued with the science of particulars1 who produces his contemplation from certain definite principles1 and runs back even to these1 but very little scrutinizes the progressions of beings1 it is requisite to attempt1 consider1 and treat concerning that impartible nature alone1 which regards his first principles2 and to behold that simplicity which presides over all the particular subjects of his knowledge8 ,in consequence of this reasoning1 therefore1 a point alone1 according to the geometric matter1 is destitute of partition2 but unity according to that which is arithmetical4 ,and the reason of a point1 however in some other respects it may be imperfect1 yet is perfect in the present science4 ,for1 indeed1 the physician also says1 that the elements of bodies are fire and water1 and things similar to these2 and as far as to these the resolution of bodies proceeds4 ,but the natural philosopher passes on to more simple elements2 and the one defines an element simple as to sense1 but the other simple as to reason2 and both of them properly as to their peculiar science4 ,we must not1 therefore1 think that the definition of a point is faulty1 nor determine it as imperfect2 for so far as pertains to the geometric matter1 and its principles1 it is sufficiently delivered4 ,this alone1 indeed1 is wanting to its completion1 that the definition does not clearly say1 that which is impartible with me is a point2 and my principle1 and that which ,i contain as most simple1 is nothing else than this4 ,and after this manner it is proper to hear the geometrician addressing us4 ,euclid1 therefore1 from a negation of parts1 declares to us a principle1 leading to the theory of its whole subject nature4 ,for negative discourses are proper to principles1 as ,parmenides teaches us1 who delivers the doctrine concerning the first and last cause1 by negations alone4 ,since every principle consists of an essence different from its flowing consequents2 and the negations of these exhibit to us the property of their source4 ,for that it is1 indeed1 the cause of these1 yet at the same time has nothing in common with these1 becomes perspicuous from a doctrine of this kind4 ,but here a doubt may arise1 how1 since the phantasy receives all things invested with forms1 and in a partible manner1 the geometrician beholds in it the point destitute of parts8 ,for it is not because they are reasons existing in cogitation1 but the phantasy receives the resemblances of intellectual and divine forms according to its own proper nature1 exhibiting in its shadowy bosom the forms of formless natures1 and clothing with figure things entirely free from the affections of figure4 ,to this ambiguity we must say1 that the species of imaginative motion is neither alone partible1 nor impartible2 but that it proceeds from the impartible to the partible1 and from the formless nature to that which is expressed by form4 ,for if it was partible alone1 it could not preserve in itself many impressions of forms1 since the subsequent would obscure the pre-existent figures3 for no body can contain at once1 and according to the same situation1 a multitude of figures2 but the former will be blotted out by the succession of the latter4 ,but if it was alone impartible1 it would not be inferior to cogitation1 and to soul1 which surveys all things in an impartible manner4 ,hence1 it is necessary that it should indeed begin from an impartible according to its motion1 and from thence draw forth the folded and scattered form of every thing falling under cogitation1 and penetrating to its shadowy receptacle3 but1 that it should at length end in form1 figure1 and interval4 ,and if it be allotted a nature of this kind1 it will1 after a certain manner1 contain an impartible essence3 and a point1 according to this1 must be said to have its principal subsistence3 for the form of a line is contracted in the phantasy according to this4 ,hence1 because it possesses a twofold power1 impartible and partible1 it will indeed contain a point in an impartible1 and intervals in a partible manner4 ,but as the ,pythagoreans define a point to be unity having position1 let us consider what they mean4 ,that numbers1 indeed1 are more immaterial and more pure than magnitudes1 and that the principle of numbers is more simple than the principle of magnitudes1 is manifest to every one3 but when they say that a point is unity endued with position1 they appear to me to evince that unity and number subsist in opinion3 ,i mean monadic number#aca4 ,on which account1 every number1 as the pentad and the heptad1 is one in every soul1 and not many2 and they are destitute of figure and adventitious form4 ,but a point openly presents itself in the phantasy1 subsists1 as it were1 in place1 and is material according to intelligible matter4 ,unity1 therefore1 has no position1 so far as it is immaterial1 and free from all interval and place3 but a point has position1 so far as it appears seated in the bosom of the phantasy1 and has a material subsistence4 ,but unity is still more simple than a point1 on account of the community of principles4 ,since a point exceeds unity according to position2 but appositions in incorporeals produce diminutions of those natures1 by which the appositions are received4 ,,definition ,,ii4 ,a ,line is a ,length without ,breadth4 ,a ,line obtains the second place in the ,definitions1 as it is by far the first and most simple interval1 which the geometrician calls a length1 adding also without breadth2 since a line1 in respect of a superficies1 ranks as a principle4 ,for he defines a point1 as it is the principle of all magnitudes1 by negation alone2 but a line1 as well by affirmation as by negation4 ,hence it is a length1 and by this exceeds the impartibility of a point2 but it is without breadth1 because it is separated from other dimensions4 ,for1 indeed1 every thing which is void of breadth1 is also destitute of bulk1 but the contrary is not true1 that every thing void of bulk is also destitute of breadth4 ,since1 therefore1 he has removed breadth from a line1 he has also removed at the same time bulk4 ,on which account he does not add1 that a line also has no thickness1 because this property is consequent to the notion of being without breadth4 ,but it is defined by others in various ways3 for some call it the flux of a point1 but others a magnitude contained by one interval4 ,and this definition1 indeed2 is perfect1 and sufficiently explains the essence of a line2 but that which calls it the flux of a point1 appears to manifest its nature from its producing cause2 and does not express every line1 but alone that which is immaterial4 ,for this is produced by a point1 which though impartible itself1 is the cause of being to partible natures4 ,but the flux of a point1 shews its progression and prolific power1 approaching to every interval1 receiving no detriment1 perpetually abiding the same1 and affording essence to all partible magnitudes4 ,however1 these observations are known1 and manifest to every one4 ,but we shall recall into our memory1 discourses more ,pythagorical1 which determine a point as analogous to unity1 a line to the duad1 a superficies to the triad1 and body to the tetrad4 #acb,yet when we compare those which receive interval together1 we shall find a line monadic2 but a superficies dyadic1 and a solid body triadic4 ,from whence also1 ,aristotle#acc says2 that body is perfected by the ternary number4 ,and1 indeed1 this is not wonderful1 that a point1 on account of its impartibility1 should be assimilated to unity2 but that things subsequent to a point1 should subsist according to numbers proceeding from unity1 and should preserve the same proportion to a point1 as numbers to unity2 and that every one should participate of its proximate superior1 and have the same proportion to its kindred1 and following degree1 as the superior to this1 which is the immediate consequent4 #acd;,for example1 that a line has the order of the duad with respect to the point1 but of unity to a superficies2 and that this last has the relation of a triad to the point1 but of the duad to a solid4 ,and on this account1 body is tetradic1 with respect to a point1 but triadic as to a line4 ,each order1 therefore1 has its proportion2 but the order of the ,pythagoreans is the more principal1 which receives its commencement from an exalted source1 and follows the nature of beings4 ,for a point is indeed twofold2 since it either subsists by itself1 or in a line2 in which last respect also1 since as a boundary it is alone and one1 neither having a whole nor parts1 it imitates the supreme nature of beings4 ,on which account too1 it was placed in a correspondent proportion to unity4 #ace;,for as the oracle says1 ,unity is there first1 where the paternal unity abides4 ,but a line is the first endued with parts and a whole1 and it is monadic because it is distant by one interval only2 and dyadic on account of its progression3 for if it be infinite1 it participates of the indefinite duad2 but if finite1 it requires two terms1 from whence and to what place2 since1 on account of these it imitates totality1 and is allotted an order among totals4 ,for unity1 according to the oracle1 is extended#acf1 and generates two2 and this produces a progression into longitude1 together with that which is distant extendedly1 and with one interval1 and the matter of the duad4 ,but superficies1 since it is both a triad and duad1 as also the receptacle of the primary figures1 and that which receives the first form and species1 is in a certain respect similar to the triadic nature1 which first terminates beings2 and to the duad1 by which they are divided and dispersed4 ,but a solid1 since it has a triple distance1 and is distinguished by the tetrad1 which is endued with a power of comprehending all reasons1 is reduced to that order in which the distinction of corporeal ornaments appears2 as also the division of the universe into three parts1 together with the tetradic property1 which is generative and female4 ,and these observations1 indeed1 might be more largely discussed1 but for the present1 must be omitted4 ,again1 the discourse of the ,pythagoreans1 not undeservedly1 calls a line1 which is the second in order1 and is constituted according to the first motion from an impartible nature1 dyadic4 ,and that a point is posterior to unity1 a line to the duad1 and a superficies to the triad1 ,parmenides himself shews1 by first of all taking away multitude from one by negation1 and afterwards the whole4 ,because1 if multitude is before that which is a whole1 number also will be prior to that which is continuous1 and the duad to the line1 and unity to the point3 since the epithet not many1 belongs to unity which generates multitude1 but to the point1 the term not a whole1 is proper1 because it produces a whole2 for this is said to have no part4 ,and these things are affirmed of a line1 while we more accurately contemplate its nature4 ,but we should also admit the followers of ,apollonius1 who say1 that we obtain a notion of a line1 when we are ordered to measure the lengths alone1 either of ways or walls2 for we do not then subjoin either breadth or bulk1 but only make one distance the object of our consideration4 ,in the same manner we perceive superficies1 when we measure fields2 and a solid1 when we take the dimensions of wells4 ,for then1 collecting all the distances together1 we say1 that the space of the well is so much1 according to length1 breadth1 and depth4 ,but a line may become the object of our sensation1 if we behold the divisions of lucid places from those which are dark1 and survey the moon when dichotomized3 for this medium has no distance with respect to latitude2 but is endued with longitude1 which is extended together with the light and shadow4 ,,definition ,,iii4 ,but the ,extremities of a ,line are ,points4 ,every composite receives its bound from that which is simple1 and every thing partible from that which is impartible2 and the images of these openly present themselves in mathematical principles4 ,for when it is said that a line is terminated by points1 it seems manifestly to make it of itself infinite1 because1 on account of its proper progression1 it has no extremity4 ,as1 therefore1 the duad is terminated by unity1 and reduces its own intolerable boldness under bound1 when it is restrained in its comprehensive embrace3 so a line also is limited by the points which it contains4 ,for1 since it is similar to the duad1 it participates of a point having the relation of unity1 according to the nature of the duad4 ,indeed1 in imaginative1 as well as in sensible forms1 the points themselves terminate the lines in which they reside4 ,but in immaterial forms1 the reason of the impartible point pre-exists separate and apart2 but when proceeding from thence by far the first of all1 by determining itself with interval1 moving itself1 and flowing in infinite progression1 and imitating the indefinite duad1 it is restrained indeed1 by its proper principle1 is united by its power1 and on every side seized by its coercive bound4 ,hence it is1 at the same time1 both infinite and finite3 infinite1 indeed1 according to its progression2 but finite according to its participation of a terminating cause4 ,so that1 when it approaches to this cause1 it is detained in its comprehension1 and is terminated according to its union4 ,hence too1 in the images of incorporeal forms1 a point is said to terminate a line1 by occupying its beginning and end4 ,bound1 therefore1 in immaterials1 is separated from that which is bounded3 but here it is twofold2 for it subsists in that which is terminated4 ,and this affords a wonderful symptom1 that forms2 indeed1 abiding in themselves1 precede their participants according to cause2 but when giving themselves up to their subordinate natures1 subsist according to their diversified properties3 since they are multiplied and distributed together with these1 and receive the division of their subjects4 ,besides1 this also must be previously received concerning a line1 that our geometrician uses it in a threefold acceptation4 ,as terminated on both sides1 and finite2 as in the problem#acg which says1 ,upon a given terminated right line to construct an equilateral triangle4 ,and as partly infinite and partly finite2 as in the problem which commands us from three right lines1 which are equal to three given right lines1 to construct a triangle2 for in the construction of the problem1 he says1 ,let there be placed a certain right line1 on one part finite1 but on the other part infinite4 ,and again1 a line is received by ,euclid as on both sides infinite2 as in the problem which says1 ,upon a given infinite right line1 from a given point1 which is not in that line1 to let fall a perpendicular4 ,but1 besides this1 the following doubts1 since they are worthy of solution1 must not be omitted4 ,how are points called the extremities of a line8 and of what line1 since they can neither be the bounds of one that is infinite1 nor of every finite8 ,for there is a certain line1 which is both finite1 and has not points for its extremities4 ,and such is a circular line1 which returns into itself1 and is not bounded by points1 like a right line4 ,and such also is the ellipsis1 or line like a shield4 ,is it therefore requisite to behold a line1 considered as a line8 for we must receive a certain circumference1 which is terminated by points1 and a part of the elliptic line2 having1 in like manner1 its extremities bounded by points4 ,but every circular and elliptic line1 assumes to itself another certain property1 by which it is not line alone1 but is also endued with a power of perfecting figure#ach4 ,lines1 themselves1 therefore1 have their extremities terminated by points2 but those which are effective of such like figures1 return into themselves4 ,and1 indeed1 if you conceive them to be described1 you will also find how they are bounded by points2 but if you receive them already described1 and connect the end with the beginning1 you can no longer behold their extremes4 ,,definition ,,iv4 ,a ,right ,line1 is that which is equally situated between its bounding ,points4 ,plato establishing two most simple and principal species of lines1 the right and the circular1 composes all the rest from the mixture of these2 ,i mean such as are called curve lines1 some of which are formed from planes2 but others subsist about solids2 and whatever species of curve lines are produced by the sections of solids4 ,and it seems1 indeed1 that a point 7if it be lawful so to speak7 bears an image of the one itself1 according to ,plato3 for unity has no part1 as he also shews in the ,parmenides4 ,but1 because after unity itself there are three hypostases1 or substances1 bound1 infinite1 and that which is mixed from these1 the species of lines1 angles1 and figures1 which subsist in the nature of things originate from thence4 ,and1 indeed1 a circumference and a circular angle1 and a circle among plane figures1 and a sphere among solids1 are analogous to bound4 ,but a right line corresponds to infinity1 according to all these2 for it properly belongs to all1 if it is beheld as existing in each4 ,but that which is mixed in all these1 is analogous to the mixt which subsists among intelligibles4 ,for lines are mixed1 as those which are called spirals4 ,and angles1 as the semi-circular and cornicular#aci4 ,and plane figures1 as segments and apsides2 but solids1 as cones and cylinders1 and others of that kind4 ,bound1 therefore1 infinite1 and that which is mixed1 are participated by all these4 ,but ,aristotle#adj likewise assents to ,plato2 for every species of lines1 says he1 is either right or circular1 or mixed from these two4 ,from whence also there are three motions1 one according to a right line2 the other circular2 and the third mixed4 ,but some oppose this division1 and say that there are not two simple lines alone1 but that there is a certain third line given1 i4 e4 a helix or spiral1 which is described about a cylinder#ada1 when1 whilst a right line is moved round the superficies of the cylinder1 a point in the line is carried along with an equal celerity4 ,for by this means1 a helix1 or circumvolute line1 is produced1 which adapts all the parts of itself to all1 according to a similitude of parts1 as ,apollonius shews in his book concerning the ,cochlea2 which passion1 among all spirals1 agrees to this alone4 ,for the parts of a plane helix are dissimilar among themselves2 as also of those which are described about a cone and sphere4 ,but the cylindric spiral alone1 consists of similar parts in the same manner as a right and circular line4 ,are there1 then1 three simple lines1 and not two only8 ,to which doubt we reply1 that a helix of this kind is1 indeed1 of similar parts1 as ,apollonius teaches1 but is by no means simple2 since among natural productions1 gold and silver are composed of similar parts1 but are not simple bodies4 ,but the generation of the cylindric helix evinces that its mixture is from things simple2 for it originates while a right line is circularly moved round the axis of the cylinder1 a point at the same time flowing along in the right line4 ,two simple motions1 therefore1 compose its nature2 and1 on this account1 it is among the number of mixt lines1 and not among such as are simple3 for that which is composed from dissimilars is not simple1 but mixt4 ,hence1 ,geminus1 with great propriety1 when he admits that some simple lines may be produced from many motions1 does not grant that every such line is mixt2 but that alone1 which arises from dissimilar motions4 ,for if you conceive a square1 and two motions which are performed with an equal celerity1 one according to the length1 but the other according to the breadth1 a right line or the diameter will be produced2 but the right line will not1 on this account1 be mixed3 for no other line precedes it1 formed by a simple motion1 as we asserted of the cylindric helix4 ,nor yet1 if you suppose a right line1 moving in a right angle1 and by a bisection to describe a circle#adb1 is the circular line1 on this account1 produced with mixture3 for the extremities of that which is moved after this manner1 since they are equally moved1 will describe a right line2 and the bisection1 since it is unequally devolved1 will delineate a circle2 but the other points will describe an ellipsis4 ,on which account1 the generation of a circular line is the consequence of that inequality of lation arising from the bisection2 because a right line was supposed to be moved in a right angle1 but not in a natural manner4 ,and thus much concerning the generation of lines4 ,but it seems1 that of the two simple lines1 the right and the circular1 the right line is the more simple2 for in this1 dissimilitude cannot be conceived1 even in opinion4 ,but in the circular line1 the concave and the convex1 indicate dissimilitude4 ,and a right line1 indeed1 does not infer a circumference according to thought2 but a circumference brings with it a right line1 though not according to its generation1 yet with respect to its centre4 ,but what if it should be said that a circumference requires a right line to its construction6 ,for if either extreme of a right line remains fixt1 but the other is moved1 it will doubtless describe a circle1 whose centre will be the abiding extreme of the right line4 ,shall we say that the generator of the circle is the point which is carried about the abiding point1 but not the right line itself8 ,for the line only determines the distance1 but the point composes the circular line1 while it is moved in a circular manner3 but of this enough4 ,again1 a circumference appears to be proximate to bound1 and to have the same proportion to other lines1 as bound to the universality of things4 ,for it is finite1 and is alone among simple lines perfective of figure4 ,but a right line is proximate to infinity2 for its capacity of infinite extension never fails3 and as all the rest are produced from bound and infinite1 in the same manner from the circular and right line1 every mixt genus of lines is composed1 as well of planes as of those which consist in solid bodies4 ,and on this account1 the soul also#adc previously assumed into herself the right and circular according to her essence1 that she might moderate all the co-ordination of infinite1 and all the nature of bound1 which the world contains4 ,by a right line1 indeed1 constituting the progression of these principles into the universe2 but by a circular line1 their return to their original source3 and by the one1 producing all things into multitude2 but by the other1 collecting them into one4 ,and not only the soul1 but he also who produced the soul1 and endued her with these powers1 contains in himself both these primary causes4 ,for when he previously assumed the beginning1 middle1 and end of all things1 he terminated right lines 7says ,plato#add71 by a circular progression according to nature4 ,and proceeding to all things by provident energies1 and returning to himself1 he established himself1 says ,timus1 after his own peculiar manner4 ,but a right line is the mark or symbol of a providence1 indeclinable1 incapable of perversion1 immaculate1 never-failing1 omnipotent1 and present to all beings1 and to every part of the universe4 ,but a circumference1 and that which environs1 is the symbol of an energy retiring into union with itself1 and which rules over all things according to one intellectual bound4 ,when1 therefore1 the demiurgus of the universe had established in himself these two principles1 the right and the circular line1 and had given them dominion1 he produced from himself two unities2 the one1 indeed1 energizing according to the circular line1 and being effective of intellectual essences2 but the other according to the right line1 and affording an origin to sensible natures4 ,but because the soul is allotted a middle situation between intellectuals and sensibles1 so far1 indeed1 as she adheres to an intellectual nature1 she energizes according to the circle2 but so far as she presides over sensibles1 she provides for their welfare according to the right line3 and thus much concerning the similitude of these forms to the universality of things4 ,but ,euclid1 indeed1 has properly delivered the present definition of a line2 by which he shews that a right line alone occupies a space equal to that which is situated between its points3 for as much as is the distance of one point from another1 so great is the magnitude of the lines terminated by the points4 ,and this is the meaning of being equally situated between its extremes4 ,for if you take two points in a circumference1 or in any other certain line1 the space of line which is included between these1 exceeds their distance from each other2 and every line1 besides a right one1 appears to suffer this property4 ,hence1 according to a common conception1 the vulgar also say1 that he who walks by a right line1 performs only a necessary journey3 but that they necessarily wander much1 who do not proceed in a right line4 ,but ,plato thus defines it2 a right line is that whose middle parts darken its extremes4 ,for this passion necessarily attends things which have a direct position2 but it is not necessary that things situated in the circumference of a circle1 or in another interval1 should be endued with this property4 ,hence1 the astrologers also say1 that the sun then suffers an eclipse when that luminary1 the moon1 and our eye are in one right line2 for it is then darkened through the middle position of the moon between us and its orb4 ,and perhaps1 the passion of a right line will evince1 that in the things which are1 according to processions emanating from causes1 the mediums are endued with a power of dividing the distance of the extremes1 and their mutual communication with each other4 ,as also1 according to regressions1 such things as are distant from the extremes1 are converted by mediums to their primary causes4 ,but ,archimedes defines a right line the least of things having the same bounds4 ,for since1 according to ,euclid1 a right line is equally situated between its points1 it is on this account1 the least of things having the same bounds3 for if a less line could be given1 it would not lie equally between its extremes3 but all the other definitions of a right line1 fall into the same conclusions2 as for instance1 that it is constituted in its extremities1 and that one part of it is not in its subject plane1 but another1 in one more sublime3 and that all its parts similarly agree to all3 and that its extremes abiding1 it also abides4 ,lastly1 that it does not perfect figure1 with one line similar in species to itself3 for all these definitions express the property of a right line1 which it possesses from the simplicity of its essence1 and from its having one progression the shortest of all from one extremity to another4 ,and thus much concerning the definitions of a right line4 ,but again1 ,geminus divides a line first into an incomposite and composite2 calling a composite1 that which is refracted1 and forms an angle2 but all the rest of them1 he denominates incomposites4 ,afterwards1 he divides a composite line into that which produces figure1 and that which may be infinitely extended4 ,and he calls that which produces figure1 a circular line1 and the line of a shield#ade1 and that which is similar to an ivy leaf#adf2 but that which is not effective of figure1 the section of a rectangular and obtuse angular cone1 the line similar to a shell#adg1 the right line1 and all of that kind4 ,and again1 after another manner1 of the incomposite line1 one sort is simple1 but the other mixt4 ,and of the simple1 one produces figure1 as the circular2 but the other is indefinite1 as the right line4 ,but of the mixt1 one subsists in planes1 but the other in solids4 ,and of that which is in planes1 one coincides in itself1 as the figure of the ivy leaf1 which is called the cissoid2 but the other may be produced in infinitum1 as the helix4 ,but of that which is in solids1 one may be considered in the sections of solids2 but the other as consisting about the solids themselves4 ,for the helix1 indeed1 which is described about a sphere or a cone1 consists about solids2 but conic1 or spirical sections are generated from a particular section of solids4 ,but1 with respect to these sections1 the conic were invented by ,mnechmus1 which also ,eratosthenes relating1 says1 8,nor in a cone ,mnechmian ternaries divide40 ,but the spiric by ,perseus1 who composed an epigram on their invention1 to this purpose1 8,when ,perseus had invented three spiral lines in five sections1 he sacrificed to the gods on the occasion40 ,and the three sections of a cone1 are the parabola1 hyperbola1 and ellipsis3 but of spiral sections1 one kind is twisted and involved1 like the fetlock of a horse2 but another is dilated in the middle1 and deficient in each extremity3 and another which is oblong1 has less space in the middle1 but is dilated on each side4 ,but the multitude of the other mixt lines is infinite4 ,for there is an innumerable multitude of solid figures1 from which there are constituted multiform sections4 ,for a right line1 while it is circularly moved#adh1 does not make a certain determinate superficies1 nor yet conical1 nor conchoidal lines1 nor circumferences themselves4 ,hence1 if these solids are multifariously cut1 they will exhibit various species of lines4 ,lastly1 of those lines which consist about solids1 some are of similar parts1 as the helixes about a cylinder2 but others of dissimilar parts1 as all the rest4 ,from these divisions1 therefore1 we may collect1 that there are only three lines of similar parts1 the right1 the circular1 and the cylindric helix4 ,the two simple ones1 indeed1 existing in a plane1 but the one mixt1 about a solid4 ,and this ,geminus evidently demonstrates1 when he shews1 that if two right lines are extended from one point1 to a line of similar parts1 so as to make equal angles upon that line1 they shall be equal to each other4 ,and the demonstrations of this may be received by the studious1 from his volumes2 since in these he delivers the origin of spiral1 conchoidal1 and cissoidal lines4 ,but we have barely related the names and divisions of these lines1 for the purpose of exciting the ingenious to their investigation2 as we think1 that an accurate enquiry after the method of detecting the properties of each1 would be superfluous in the present undertaking3 since the geometrician only unfolds to us in this work1 simple and primary lines1 i4e4 the right line1 in the present definition2 but a circular line1 in the tradition of a circle4 ,for he then says1 that the line terminating the circle1 is the circumference4 ,but he makes no mention of mixt lines1 though he was well acquainted with mixt angles1 ,i mean1 the semi-circular and cornicular3 as also with plane mixt figures1 i4e4 segments and sectors2 and with solids1 viz4 cones and cylinders4 ,of each of the rest1 therefore1 he delivers three species2 but of lines only two1 i4 e4 the right and circular3 for he thought it requisite in discourses concerning things simple1 to assume simple species2 and all the rest are more composite than lines4 ,hence1 in imitation of the geometrician1 we also shall terminate their explanation with simple lines4 ,,definition ,v4 ,a ,,superficies is that which has only ,length and ,breadth4 ,after a point and a line1 a superficies is placed1 which is distant by a twofold interval1 length and breadth4 ,but this also remaining destitute of thickness or bulk1 possesses a nature more simple than body1 which is distant by a triple dimension4 ,on which account the geometrician adds to the two intervals the particle only1 because the third interval does not exist in superficies4 ,and this is equivalent to a negation of bulk1 as here also he shews the excellency of superficies compared to a solid with respect to simplicity1 by negation1 or by an addition equivalent to negation3 but the diminution which it possesses1 if compared with the preceding terms1 by the affirmations themselves4 ,but others define a superficies to be the boundary of body1 which is almost affirming the same as the definition of ,euclid2 since that which terminates is exceeded in one dimension1 by that which is terminated4 ,and others1 a magnitude different by two intervals4 ,lastly1 others declaring the same affection1 form its assignation in a somewhat different manner4 ,but they say we have a knowledge of superficies when we measure fields1 and distinguish their extremities according to length and breadth2 but that we receive a certain sensation of it1 when we behold shadows4 ,for as they are without bulk1 because they cannot penetrate into the interior part of the earth1 they have only length and breadth4 ,but the ,pythagoreans say1 that it is assimilated to the triad2 because the ternary is by far the first cause to all the figures2 which a superficies contains4 ,for a circle1 which is the principle of orbicular figures1 occultly possesses the ternary1 by its centre1 interval1 and circumference4 ,but a triangle1 which ranks as the first among all right-lined figures1 on every side evinces that it is enclosed by the triad1 and receives its form from its perfect nature4 ,,definition ,,vi4 ,the ,extremities of a ,,superficies are ,lines4 ,from these also1 as images1 we may understand1 that things more simple procure bound and an end to every one of their proximate natures3 for soul perfects and determines the operations of nature2 and nature the motion of bodies4 ,and prior to these1 intellect measures the convolutions of soul2 and unity the life of intellect2 for that is the measure of all4 ,just as in these also1 a solid is terminated by a superficies2 but a superficies by a line2 and a line by a point2 for that is the boundary of them all4 ,hence1 the line existing uniformly in immaterial forms and impartible reasons1 terminates and restrains the various motion of a superficies in its progression1 and proximately unites its infinity4 ,but in the images of these1 when that which bounds supervenes that which is bounded1 it causes1 by this means1 its limitation and bound4 ,but if it should be enquired how lines are the extremities of every superficies1 since they are not the extremes of every finite figure2 for the superficies of a sphere is terminated indeed1 yet not by lines1 but by itself8 ,in answer to this1 we must say1 that by receiving a superficies so far as it is distant by a twofold interval1 we shall find it terminated by lines according to length and breadth4 ,but if we behold a spherical superficies1 we must receive it as that which is endued with figure2 which possesses another quality1 and conjoins the end with the beginning2 and loses its two extremities in the comprehensive embraces of one3 and this one extremity subsists in capacity only1 and not in energy4 ,,definition ,,vii4 ,a ,plane ,,superficies is that which is equally situated between its bounding ,lines4 ,it was not agreeable to the ancient philosophers to establish a plane species of superficies2 but they considered superficies in general1 as the representative of magnitude1 which is distant by a twofold interval4 ,for thus the divine ,plato#adi says1 that geometry is contemplative of planes1 opposing it in division to stereometry1 as if a plane and a superficies were the same4 ,and this was likewise the opinion of the demoniacal ,aristotle#aej4 ,but ,euclid and his followers consider superficies as a genus1 but a plane as its species1 in the same manner as rectitude of a line4 ,and on this account he defines a plane separate from a superficies1 after the similitude of a right line4 ,for he defines this last as equal to the space1 placed between its points4 ,and in like manner1 he says1 that two right lines being given1 a plane superficies occupies a place equal to the space situated between those two lines4 ,for this is equally situated between its lines2 and others also explaining the same boundary1 assert that it is constituted in its extremities4 ,but others define it as that to all the parts of which a right line may be adapted#aea4 ,but perhaps others will say1 that it is the shortest of superficies1 having the same boundaries2 and that its middle parts darken its extremities2 and that all the definitions of a right line may be transferred into a plane superficies1 by only changing the genus3 since a right1 circular1 and mixt line1 commencing from lines1 arrive even at solids1 as we have asserted above2 for they are proportionally1 both in superficies and solids4 ,hence also1 ,parmenides says1 that every figure is either right1 or circular1 or mixt4 ,but if you wish to consider the right in superficies1 take a plane1 to which a right line agrees in various ways2 but if a circular receive a spherical superficies2 and if a mixt1 a conic or cylindric1 or some one of that genus4 ,but it is requisite 7says ,geminus7 since a line1 and also a superficies is called mixt1 to know the measure of mixture1 because it is various4 ,for mixture in lines1 is neither by composition1 nor by temperament only3 since1 indeed1 a helix is mixed1 yet one part of it is not straight1 and another part circular1 like those things which are mixed by composition3 nor if a helix is cut after any manner1 does it exhibit an image of things simple1 such as those which are mixed through temperament2 but in these the extremes are1 at the same time1 corrupted and confused4 ,hence1 ,theodorus the mathematician1 does not rightly perceive1 in thinking that this mixture is in lines4 ,but mixture in superficies1 is neither by composition1 nor by confusion2 but subsists rather by a certain temperament4 ,for conceiving a circle in a subject plane1 and a point on high1 and producing a right line from the point to the circumference of the circle1 the revolution of this line will produce a conical superficies which is mixt4 ,and we again resolve it into its simple elements1 by a parallel section3 for by drawing a section between the vertex and the base1 which shall cut the plane of the generative right line1 we effect a circular line4 ,but the idea of lines1 shews that the mode of mixture is not by temperament2 for neither does it send us back to the simple nature of elements3 on the contrary1 when superficies are cut1 they immediately exhibit to us their producing lines4 ,the mode of mixture1 therefore1 is not the same in lines and superficies4 ,but as among lines there were some simple1 that is1 the right and circular1 of which the vulgar also possess an anticipated knowledge without any previous instruction2 but the species of mixt lines require a more artificial apprehension3 so among superficies1 we possess an innate notion of those which are especially elementary1 the plane and spherical2 but science and its reason investigates the variety of those which are composed through mixture4 ,but this is an admirable property of superficies1 that their mixture in generation is oftentimes produced from a circular line2 and this also happens to a spiral superficies4 ,for this is understood by the revolution of a circle remaining erect1 and turning itself about the same point which is not its centre4 ,and on this account1 a spiral also is threefold2 for its centre is either in a circumference1 or within1 or external to a circumference4 ,if the centre is in the circumference1 a continued spiral is produced3 if within the circumference1 an intangled one2 if without1 a divided one4 ,and there are three spiral sections corresponding to these three differences4 ,but every spiral line is mixt1 although the motion from which it is produced is one and circular4 ,and mixt superficies are produced as well from simple lines1 7as we have said17 while they are moved with a motion of this kind1 as from mixt lines4 ,since1 therefore1 there are three conic lines1 they produce four mixt superficies1 which they call conoids4 ,for a rectangular conoid1 is produced from the revolution of the parabola about its axis3 but that which is formed by the ellipsis1 is called a spheroid2 and is the revolution is made about the greater axis1 it is an oblong2 but if about the lesser a broad spheroid4 ,lastly1 an obtuse-angled conoid is generated from the revolution of the hyperbola4 ,but it is requisite to know1 that sometimes we arrive at the knowledge of superficies from lines1 and sometimes the contrary2 for from conical and spiral superficies1 we apprehend conical and spiral lines4 ,besides1 this also must be previously received concerning the difference of lines and superficies1 that there are three lines of similar parts 7as we have already observed71 but only two superficies1 the plane and the spherical4 ,for this is not true of the cylindric1 since all parts of the cylindric superficies cannot agree to all4 ,and thus much concerning the differences of superficies1 one of which the geometrician having chosen 7,i mean the plane71 this also he has defined2 and in this1 as a subject1 he contemplates figures1 and their attendant passions3 for his discourse is more copious in this than in other superficies3 since1 indeed1 we may understand right lines1 and circles1 and helixes in a plane2 also the sections of circles and right lines1 contacts1 and applications1 and the constructions of angles of every kind4 ,but in other superficies1 all these cannot be beheld4 ,for how in one that is spherical1 can we apprehend a right line1 or a right-lined angle8 ,how1 lastly1 in a conic or cylindric superficies1 can we behold sections of circles or right lines8 ,not undeservedly1 therefore1 does he both define this superficies1 and discuss his geometrical concerns1 by exhibiting every thing in this as in a subject2 for from hence he calls the present treatise plane4 ,and1 after this manner1 it is requisite to understand that which is plane1 as projected and constituted before the eyes3 but cogitation as describing all things in this1 the phantasy corresponding to a plane mirror1 and the reasons resident in cogitation as dropping their images#aeb into its shadowy receptacle4 ,,definition ,,viii4 #aec;,a ,,plane ,,angle1 is the inclination of two ,lines to each other in a ,plane1 which meet together1 but are not in the same direction4 ,some of the ancient philosophers1 placing an angle in the predicament of relation1 have said1 that it is the mutual inclination of lines or planes to each other4 ,but others1 including this in quality1 as well as rectitude and obliquity1 say1 that it is a certain passion of a superficies or a solid4 ,and others1 referring it to quantity1 confess that it is superficies or a solid4 ,for the angle which subsists in superficies is divided by a line2 but that which is in solids1 by a superficies4 ,but 7say they7 that which is divided by these1 is no other than magnitude1 and this is not linear1 since a line is divided by a point2 and therefore it follows that it must be either a superficies or a solid4 ,but if it is magnitude1 and all finite magnitudes of the same kind have a mutual proportion2 all angles of the same kind1 i4 e4 which subsist in superficies1 will have a mutual proportion4 ,and hence1 the cornicular will be proportionable to a right-lined angle4 ,but things which have a mutual proportion1 may1 by multiplication1 exceed each other2 and therefore it may be possible for the cornicular to exceed a right-lined angle1 which1 it is well known1 is impossible1 since it is shewn to be less than every right-lined angle4 ,but if it is quality alone1 like heat and cold1 how is it divisible into equal parts8 ,for equality1 inequality1 and divisibility1 are not less resident in angles than in magnitudes2 but they are1 in like manner1 essential4 ,but if the things in which these are essentially inherent1 are quantities1 and not qualities1 it is manifest that angles also are not qualities4 ,since the more and the less are the proper passions of quality#aed1 but not equal and unequal4 ,on this hypothesis1 therefore1 angles ought not to be called unequal1 and this greater1 but the other less2 but they ought to be denominated dissimilars1 and one more an angle1 but the other less4 ,but that these appellations are foreign from the essence of mathematical concerns1 is obvious to every one3 for every angle receives the same definition1 nor is this more an angle1 but that less4 ,thirdly1 if an angle is inclination1 and belongs to the category of relation1 it must follow1 that from the existence of one inclination1 there will also be one angle1 and not more than one4 ,for if it is nothing else than the relation of lines or planes1 how is it possible there can be one relation of lines or planes1 but many angles8 ,if1 therefore1 we conceive a cone cut by a triangle from the vertex to the base1 we shall behold one inclination of the triangular lines in the semicone to the vertex2 but two distinct angles3 one of which is plane1 ,i mean that of the triangle2 but the other subsists in the mixt superficies of the cone1 and both are comprehended by the two triangular lines4 ,the relation1 therefore1 of these1 do not make the angle4 ,again1 if is necessary to call an angle either quality or quantity1 or relation2 for figures1 indeed1 are qualities1 but their mutual proportions belong to relation4 ,it is necessary1 therefore1 that an angle should be reduced under one of these three genera4 ,such doubts1 then1 arising concerning an angle1 and ,euclid calling it inclination1 but ,apollonius the collection of a superficies1 or a solid in one point1 under a refracted line or superficies 7for he seems to define every angle universally71 we shall affirm1 agreeable to the sentiments of our preceptor ,syrianus1 that an angle is of itself none of the aforesaid2 but is constituted from the concurrence of them all4 ,and that1 on this account1 a doubt arises among those who regard one category alone4 ,but this is not peculiar to an angle1 but is likewise the property of a triangle4 ,for this1 too1 participates of quantity1 and is called equal and unequal2 because it has to quantity the proportion of matter4 ,but quality also1 is present with this1 in consequence of its figure 7since triangles are called as well similar as equal72 but it possesses this from one category1 and that from another4 ,hence1 an angle is perfectly indigent of quantity1 the subject of magnitude4 ,but it is also indigent of quality1 by which it possesses1 as it were1 its proper form and figure1 ,lastly1 it is indigent of the relation of lines terminating1 or of superficies comprehending its form4 ,so that an angle consists from all these1 yet is not any one of them in particular4 ,and it is indeed divisible1 and capable of receiving equality and inequality1 according to the quantity which it contains4 ,but it is not compelled to admit the proportion of magnitudes of the same kind1 since it has also a peculiar quantity1 by which angles are also incapable of a comparison with each other4 ,nor can one inclination perfect one angle3 since the quantity also1 which is placed between the inclined lines1 completes its essence4 ,if then we regard these distinctions1 we shall dissolve all absurdities1 and discover that the property of an angle is not the collection of a superficies or solid1 according to ,apollonius 7since these also complete its essence17 but that it is nothing else than a superficies itself1 collected into one point1 and comprehended by inclined lines1 or by one line inclined to itself3 and that a solid angle is the collection of superficies mutually inclined to each other4 ,hence1 we shall find that a formed quantum1 constituted in a certain relation1 supplies its perfect definition4 ,and thus much we have thought requisite to assert concerning the substance of angles1 previously contemplating the common essence of every triangle1 before we divide it into species4 ,but since there are three opinions of an angle1 ,eudemus the ,peripatetic1 who composed a book concerning an angle1 affirms that it is quality4 ,for1 considering the origin of an angle1 he says that it is nothing else than the fraction of lines3 because1 if rectitude is quality1 fraction also will be quality4 ,and hence1 since its generation is in quality1 an angle will be entirely quality4 ,but ,euclid1 and those who call it inclination1 place it in the category of relation4 ,but they call it quantity1 who say that it is the first interval under a point1 that is immediately subsisting after a point4 ,in the number of which is ,plutarch1 who constrains ,apollonius also into the same opinion4 ,for it is requisite 7says he7 there should be some first interval1 under the inclination of containing lines or superficies4 ,but since the interval1 which is under a point1 is continuous1 it is not possible that a first interval can be assumed2 since every interval is divisible in infinitum4 ,besides1 if we any how distinguish a first interval1 and through it draw a right line1 a triangle is produced1 and not one angle4 ,but ,carpus ,antiochenus says1 that an angle is quantity1 and is the distance of its comprehending lines1 or superficies2 and that this is distant by one interval1 and yet an angle is not on that account a line3 since it is not true that every thing which is distant by only one interval1 is a line4 ,but this surely is the most absurd of all1 that there should be any magnitude except a line1 which is distant only by one interval4 ,and thus much concerning the nature of an angle4 ,but with respect to the division of angles1 some consist in superficies1 but others in solids4 ,and of those which are in superficies1 some are in simple ones1 but others in such as are mixt4 ,for an angle may be produced in a cylindric1 conic1 spherical1 and plane superficies4 ,but of those which consist in simple superficies1 some are constituted in the spherical2 but others in the plane4 ,for the zodiac itself forms angles1 dividing the equinoctial in two parts1 at the vertex of the cutting superficies4 ,and angles of this kind subsist in a spherical superficies4 ,but of those which are in planes1 some are comprehended by simple lines1 others by mixt ones2 and others1 again1 by both4 ,for in the shield-like figure#aee1 an angle is comprehended by the axis1 and the line of the shield3 but one of these lines is mixt1 and the other simple4 ,but if a circle cuts the shield1 the angle will be comprehended by the circumference1 and the ellipsis4 ,and when cissoids1 or lines similar to an ivy leaf1 closing in one point like the leaves of ivy 7from whence they derive their appellation7 make an angle1 such an angle is comprehended by mixt lines4 ,also1 when the hippopede1 or line familiar to the foot of a mare1 which is one of the spirals1 inclining to another line1 forms an angle1 it is comprehended by mixt lines4 ,lastly1 the angles contained by a circumference and a right line1 are comprehended by simple lines4 ,but of these again1 some are contained by such as are similar in species1 but others by such as are dissimilar4 ,for two circumferences1 mutually cutting1 or touching each other1 produce angles3 and these triple1 for they are either on both sides convex1 when the convexities of the circumferences are external3 or on both sides concave1 when both the concavities are external2 which they call sistroides2 or mixt from convex and concave lines1 as the lines called lunulas4 ,but besides this1 angles are contained in a twofold manner1 by a right line and a circumference3 for they are either contained by a right line1 and a concave circumference1 as the semi-circular angle2 or by a right line and a convex circumference1 as the cornicular angle4 ,but all those which are comprehended by two right lines1 are called rectilinear angles1 which have likewise a triple difference#aef4 ,the geometrician1 therefore1 in the present hypothesis1 defines all those angles which are constituted in plane superficies1 and gives them the common name of a plane angle4 ,and the genus of these he denominates inclination3 but the place1 the plane itself1 for angles have position3 but their origin such1 that it is requisite there should be two lines at least1 and not three as in a solid4 ,and that these should touch each other1 and by touching1 must not lie in a right line1 as an angle is the inclination and comprehension of lines3 but is not distance only1 according to one interval4 ,but if we examine this definition1 in the first place it appears that it does not admit1 an angle can be perfected by one line2 though a cissoid1 which is but one1 perfects an angle4 ,and1 in like manner1 the hippopede4 ,for we call the whole a cissoid1 and not its portions 7lest any one should say1 that the conjunction of these forms an angle7 and the whole a spiral1 but not its parts4 ,each1 therefore1 since it is one1 forms an angle to itself1 and not to another4 ,but after this1 he is faulty1 in defining an angle to be inclination4 ,for how1 on this hypothesis1 will there be two angles1 from one inclination8 ,how can we call angles equal and unequal8 ,and whatever else is usually objected against this opinion4 ,thirdly1 and lastly1 that part of the definition1 which says1 and not placed in a right line1 is superfluous in certain angles1 as in those which are formed from orbicular lines4 ,for without the assistance of this part1 the definition is perfect2 since the inclination of one of the lines to the other1 forms the angle4 ,and it is not possible that orbicular angles should be placed in a right line4 ,and thus much we have thought proper to say concerning the definition of ,euclid2 partly1 indeed1 interpreting1 and partly doubting its truth4 ,,definition ,,ix4 ,but when the ,lines containing the ,angle1 are right1 the ,angle is called ,,rectilinear4 ,an angle is the symbol and image of the connection and compression1 which subsists in the divine genera1 and of that order which collects divisibles into one1 particles into an impartible nature1 and the many into conciliating community4 ,for it is the bond of a multitude of lines and superficies1 the collector of magnitude into the impartibility of points1 and the comprehender of every figure which is composed by its confining nature4 ,on which account1 the oracles#aeg call the angular junctions of figures1 knots1 so far as they bring with them an image of connecting union1 and divine conjunctions1 by which discrete natures mutually cohere with each other4 ,the angles1 therefore1 subsisting in superficies1 express the more immaterial1 simple1 and perfect unions which superficies contain3 but those which are in solids1 represent the unions1 which proceed even to inferiors1 and supply a community to things disjunct1 and a construction of the same nature1 to things which on every side receive a perfect partition4 ,but of the angles in superficies1 some shadow forth primary and unmixt unions2 but others1 such as comprehend in themselves1 an infinity of progressions4 ,and some1 indeed1 are the sources of union to intellectual forms2 but others1 to sensible reasons2 and others1 again1 are copulative of those forms which obtain between these1 a middle situation4 ,hence1 the angles which are made from circumferences1 imitate those causes which envelop intellectual variety in coercive union2 for circumferences1 hastening to coalesce with each other1 are images of intellect1 and intellectual forms4 ,on the contrary1 rectilineal angles1 are the symbols of those unions which preside over sensibles1 and afford a conjunction of the reasons subsisting in these3 but mixt angles represent the preservers of the communion1 as well of sensible1 as of intellectual forms1 according to one immoveable union4 ,it is requisite1 therefore1 by regarding these paradigms1 or exemplars1 to render the causes of each4 ,for among the ,pythagoreans we shall find various angles dedicated to various gods4 ,thus1 ,philolaus1 consecrates to some a triangular1 but to others a quadrangular angle2 and to others1 again1 different angles4 ,likewise1 he permits the same to many gods1 and many to the same god1 according to the different powers which they contain4 ,and with a view to this1 and to the demiurgic triangle1 which is the primary cause of all the ornament of the elements1 it appears to me1 that ,theodorus ,asinus the philosopher1 constitutes some of the gods1 according to sides2 but others1 according to angles4 ,the first1 indeed1 supplying progression and power2 but the second1 the conjunction of the universe1 and the collection of progressive natures again into one4 ,but these1 indeed1 direct us to the knowledge of the things which are4 ,and we must not wonder that lines are here said to contain an angle4 ,for the one and impartible nature which is found in these1 is adventitious3 but in the gods themselves1 and in true beings1 the whole1 and impartible good1 precedes things many1 and divided4 ,,definition ,x4 ,when a ,right ,line standing on a ,right ,line1 makes the successive ,angles on each side equal to one another1 each of the equal ,angles1 is a ,right ,angle2 and the insisting ,right ,line1 is called a ,,perpendicular to that upon which it stands4 ,,definition ,,xi4 ,an ,,obtuse ,,angle is that which is greater than a ,,right ,,angle4 ,,definition ,,xii4 ,but an ,,acute ,,angle1 is that which is less than a ,,right ,,angle4 ,these are the triple species of angles1 which ,socrates speaks of in the ,republic1 and which are received by geometricians from hypothesis2 a right-line constituting these angles1 according to a division into species2 ,i mean1 the right1 the obtuse1 and the acute4 ,the first of these being defined by equality1 identity and similitude2 but the others being composed through the nature of the greater and the lesser2 and lastly1 through inequality and diversity1 and through the more and the less1 indeterminately assumed4 ,but many geometricians1 are unable to render a reason of this division1 and use the assertion1 that there are three angles1 as an hypothesis#aeh4 ,so that1 when we interrogate them concerning its cause1 they answer1 this is not to be required of them as geometricians4 ,however1 the ,pythagoreans1 referring the solution of this triple distribution to principles1 are not wanting in rendering the causes of this difference of right-lined angles4 ,for1 since one of the principles subsists according to bound1 and is the cause of limitation1 identity1 and equality1 and lastly1 of the whole of a better co-ordination3 but the other is of an infinite nature1 and confers on its progeny1 a progression to infinity1 increase1 and decrease1 inequality1 and diversity of every kind1 and entirely presides over the worse series2 hence1 with great propriety1 since the principles of a right-lined angle are constituted by these1 the reason proceeding from bound1 produces a right angle1 one1 with respect to the equality of every right angle1 endued with similitude1 always finite and determinate1 ever abiding the same1 and neither receiving increment nor decrease4 ,but the reason proceeding from infinity1 since it is the second in order1 and of a dyadic nature1 produces twofold angles about the right angle1 distinguished by inequality1 according to the nature of the greater and the lesser1 and possessing an infinite motion1 according to the more and the less1 since the one becomes more or less obtuse2 but the other more or less acute4 ,hence1 in consequence of this reason1 they ascribe right angles to the pure and immaculate gods of the divine ornaments1 and divine powers which proceed into the universe1 as the authors of the invariable providence of inferiors2 for rectitude1 and an inflexibility and immutability to subordinate natures accords with these gods3 but they affirm1 that the obtuse and acute angles should be ascribed to the gods1 who afford progression1 and motion1 and a variety of powers4 ,since obtuseness is the image of an expanded progression of forms2 but acuteness possesses a similitude to the cause dividing and moving the universe4 ,but likewise1 among the things which are1 rectitude is1 indeed1 similar to essence1 preserving the same bound of its being2 but the obtuse and acute1 shadow forth the nature of accidents4 ,for these receive the more and the less1 and are indefinitely changed without ceasing4 ,hence1 with great propriety1 they exhort the soul to make her descent into generation1 according to this invariable species of the right angle1 by not verging to this part more than to that2 and by not affecting some things more1 and others less4 ,for the distribution of a certain convenience and sympathy of nature1 draws it down into material error1 and indefinite variety#aei4 ,a perpendicular line is1 therefore1 the symbol of inflexibility1 purity1 immaculate1 and invariable power1 and every thing of this kind4 ,but it is likewise the symbol of divine and intellectual measure3 since we measure the altitudes of figures by a perpendicular1 and define other rectilineal angles by their relation to a right angle1 as by themselves they are indefinite and indeterminate4 ,for they are beheld subsisting in excess and defect1 each of which is1 by itself1 indefinite4 ,hence they say1 that virtue also stands according to rectitude2 but that vice subsists according to the infinity of the obtuse and acute1 that it produces excesses and defects1 and that the more and the less exhibit its immoderation1 and inordinate nature4 ,of rectilineal angles1 therefore1 we must establish the right angle1 as the image of perfection1 and invariable energy1 of limitation1 intellectual bound1 and the like2 but the obtuse and acute1 as shadowing forth infinite motion1 unceasing progression1 division1 partition and infinity4 ,and thus much for the theological speculation of angles4 ,but here we must take notice1 that the genus is to be added to the definitions of an obtuse and acute angle2 for each is right-lined1 and the one is greater1 but the other less than a right-angle4 ,but it is not absolutely true1 that every angle which is less than a right one1 is acute4 ,for the cornicular is less than every right-angle1 because less than an acute one1 yet is not on this account an acute angle4 ,also1 a semi-circular is less than any right-angle1 yet is not acute4 ,and the cause of this property is because they are mixt1 and not rectilineal angles4 ,besides1 many curve-lined angles appear greater than right-lined angles1 yet are not on this account obtuse2 because it is requisite that an obtuse should be a right-lined angle4 ,secondly1 as it was the intention of ,euclid1 to define a right-angle1 he considers a right-line standing upon another right-line1 and making the angles on each side equal4 ,but he defines an obtuse and acute angle1 not from the inclination of a right line to either part1 but from their relation to a right-angle4 ,for this is the measure of angles deviating from the right1 in the same manner as equality of things unequal4 ,but lines inclined to either part1 are innumerable1 and not one alone1 like a perpendicular4 ,but after this1 when he says1 7the angles equal to one another7 he exhibits to us a specimen of the greatest geometrical diligence2 since it is possible that angles may be equal to others1 without being right4 ,but when they are equal to one another1 it is necessary they should be right4 ,besides1 the word successive appears to me not to be added superfluously1 as some have improperly considered it2 since it exhibits the reason of rectitude4 ,for it is on this account that each of the angles is right2 because1 when they are successive1 they are equal4 ,and1 indeed1 the insisting right-line1 on account of its inflexibility to either part1 is the cause of equality to both1 and of rectitude to each4 ,the cause1 therefore1 of the rectitude of angles1 is not absolutely mutual equality1 but position in a consequent order1 together with equality4 ,but1 besides all this1 ,i think it here necessary to call to mind1 the purpose of our author2 ,i mean1 that he discourses in this place1 concerning the angles consisting in one plane4 ,and hence1 this definition is not of every perpendicular2 but of that which is in one and the same plane4 ,for it is not his present design to define a solid angle4 ,as1 therefore1 he defines1 in this place1 a plane angle1 so likewise a perpendicular of this kind4 ,because a solid perpendicular ought not to make right angles to one right-line only2 but to all which touch it1 and are contained in its subject plane3 for this is its necessary peculiarity4 ,,definition ,,xiii4 ,a ,,bound is that which is the ,extremity of any thing#afj4 ,a ,bound1 in this place1 is not to be referred to all magnitudes1 for there is a bound and extremity of a line2 but to the spaces which are contained in superficies1 and to solid bodies4 ,for he now calls a bound1 the ambit which terminates and distinguishes every space4 ,and a bound of this kind1 he defines to be an extremity3 but not after the manner in which a point is called the extremity of a line1 but according to its property of including and excluding from circumjacent figures4 ,but this name is proper to geometry in its infant state1 by which they measured fields1 and preserved their boundaries distinct and without confusion1 and from which they arrived at the knowledge of the present science4 ,since1 therefore1 ,euclid calls the external ambit1 a bound1 it is not without propriety that he1 by this means1 defines the extremity of spaces4 ,for by this1 every thing comprehended is circumscribed4 ,i say1 for example1 in a circle1 its bound and extremity is the circumference2 but itself1 a certain plane space3 and so of the rest4 ,,definition ,,xiv4 ,a ,,figure is that which is comprehended by one or more ,boundaries4 ,because figure is predicated in various ways1 and is divided into different species1 it is requisite1 in the first place1 to behold its differences2 and afterwards to discourse concerning that figure which is proposed in this ,definition4 ,there is1 then a certain figure which is constituted by mutation1 and is produced from passion1 while the recipients of the figure are disturbed1 divided1 or taken away2 while they receive additions1 or are altered1 or suffer other various affections4 ,there is also a figure1 which is produced by the potter's1 or statuary's art1 according to the pre-existent reason1 which art itself contains3 art1 indeed1 producing the form1 but matter receiving from thence1 form1 and beauty1 and elegance4 ,but there are still more noble and more illustrious figures than these1 the skilful operations of nature4 ,some1 indeed1 existing in the elements under the moon#afa1 and having a power of comprehending the reasons those elements contain3 but others are situated in the celestial regions1 distinguishing their powers1 and endless revolutions4 ,for the heavenly bodies1 both when considered by themselves1 and with relation to each other1 exhibit an abundant and admirable variety of figures2 and at different times they present to our view different forms1 bringing with them a splendid image of intellectual species2 and1 by their elegant and harmonious revolutions1 describing the incorporeal and immaterial powers of figures4 ,but there are1 again1 besides all these1 most pure and perfect beauties1 the figures of souls1 which1 because they are full of life1 and self-motive1 have an existence prior to things moved by another2 and which1 because they subsist immaterially1 and without any dimension1 excel the forms which are endued with dimension and matter4 ,in the nature of which we are instructed by ,timus1 who has explained to us the demiurgic1 and essential figure of souls#afb4 ,but again1 the figures of intellects are by far more divine than the figures of souls2 for these1 on every side1 excel partible essences2 are every where resplendent with impartible and intellectual light2 are prolific1 effective1 and perfective of the universe2 are equally present1 and firmly abide in all things2 and procure union to the figures of souls2 but recall the mutation of sensible figures to the limitation of their proper bound4 ,lastly1 there are1 separate from all these1 those perfect1 uniform1 unknown1 and ineffable figures of the gods1 which are resident1 indeed1 in the figures of intellects2 but jointly terminate all figures1 and comprehend all things in their unifying boundaries4 ,the properties of which the theurgic art1 also expressing1 surrounds various resemblances of the gods1 with various figures4 ,and some1 indeed1 it fashions by characters1 in an ineffable manner2 for characters of this kind1 manifest the unknown powers of the gods3 but others it imitates by forms and images2 fashioning some of them erect1 and others fitting2 and some similar to a heart1 but others spherical1 and others expressed by different figures4 ,and again1 some it fabricates of a simple form2 but others it composes from a multitude of forms2 and some are sacred and venerable2 but others are domestic1 exhibiting the peculiar gentleness of the gods4 ,and some it constructs of a severe aspect2 and lastly1 attributes to others1 different symbols1 according to the similitude and sympathy pertaining to the gods#afc4 ,since1 therefore1 figure derives its origin from the gods themselves1 it arrives1 by a gradual progression1 even to inferiors1 in these also appearing from primary causes4 ,since it is requisite to suppose the perfect before the imperfect1 and things situated in the stability of their own essence1 prior to those which subsist in others1 and previous to things full of their own privation1 such as preserve their proper nature sincere4 ,such figures1 therefore1 as are material1 participate of material inelegance1 and do not possess a purity convenient to their nature4 ,but the celestial figures are divisible1 and subsist in others4 ,and the figures of souls are endued with division1 and variety1 and involution of every kind2 but the figures of intellects1 together with immaterial union1 possess a progression into multitude4 ,and lastly1 the figures of the gods are free1 uniform1 simple1 and generative2 they subsist before all things1 containing all perfection in themselves1 and extending from themselves to all things1 the completion of forms4 ,we must not1 therefore1 listen to1 and endure the opinions of many1 who affirm1 that certain additions1 ablations1 and alterations1 produce sensible figures1 7for motions1 since they are imperfect1 cannot possess the principle and primary cause of effects2 nor could the same figures often be produced from contrary motions2 for the same form is sometimes generated from addition and detraction17 but we must consider operations of this kind as subservient to other purposes in generation1 and derive the perfection of figure from other primogenial causes4 ,nor must we subscribe to their opinion1 who assert that figures destitute of matter can have no subsistence2 but those only which appear in matter4 ,nor to theirs1 who acknowledge1 indeed1 that they are external to matter1 but consider them as subsisting alone1 according to thought and abstraction4 ,for where shall we preserve in safety1 the certainty1 beauty1 and order of figures1 among things which subsist by abstraction8 ,for1 since they are of the same kind with sensibles1 they are far distant from indubitable and pure certainty4 ,but from whence do they derive the certainty1 order1 and perfection which they receive8 ,for they either derive it from sensibles 7but they have no subsistence in these71 or from intelligibles 7but in these they are more perfect71 since1 to say from that which is not1 is the most absurd of all4 ,for nature does not produce imperfect figures1 and leave the perfect without any subsistence4 ,nor is it lawful1 that our soul should fabricate more certain1 perfect1 and orderly figures1 than intellect and the gods themselves4 ,there are1 therefore1 prior to sensible figures1 self-moving1 intellectual1 and divine reasons of figures4 ,and we are excited1 indeed1 from the obscurity of sensible forms1 but we produce internal reasons1 which are the lucid images of others4 ,and we possess a knowledge of sensible figures1 by their exemplars resident in soul 771 but we comprehend by images 77 such as are intellectual and divine4 ,for the reasons we contain1 emerging from the dark night of oblivion1 and propagating themselves in sciential variety1 exhibit the forms of the gods1 and the uniform bounds of the universe1 by which they ineffably convert all things into themselves4 ,in the gods1 therefore1 there is both an egregious knowledge of universal figures1 and a power of generating and constituting all inferiors4 ,but in natures1 figures are endued with a power generative of apparent forms2 but are destitute of cognition and intellectual perception4 ,and1 in particular souls1 there is1 indeed1 an immaterial intellection1 and a self-energizing knowledge2 but there is wanting a prolific1 and efficacious cause4 ,as1 therefore1 nature1 by her forming power presides over sensible figures1 in the same manner1 soul1 by her gnostic energy1 drops in the phantasy as in a mirror1 the reasons of figures4 ,but the phantasy receiving these in her shadowy forms1 and possessing images of the inherent reasons of the soul1 affords by these the means of inward conversion to the soul1 and of an energy directed to herself1 from the spectres of imagination4 ,just as if any one beholding his image in a mirror1 and admiring the power of nature1 and his own beauty1 should desire to see himself in perfection1 and should receive a power of becoming1 at the same time1 the perceiver1 and the thing perceived4 ,for the soul1 after this manner1 looking abroad into the bright mirror of the phantasy1 and surveying the shadowy figures it contains1 and admiring their beauty and order1 pursues1 in consequence of her admiration1 the reasons from which these images proceed2 and being wonderfully delighted1 dismisses their beauty1 as conversant about spectres alone2 but afterwards seeks her own purer beauty1 and desires to pass into her own profound retreats1 and there to perceive the circle and the triangle1 and all things subsisting together1 in an impartible manner1 and to insert herself in the objects1 to contract her multitude into one2 and lastly1 to behold the occult and ineffable figures of the gods1 seated in the most sacred and divine recesses of her nature4 ,she is likewise desirous of bringing into light1 from its awful concealment1 the solitary beauty1 of the gods1 and of perceiving the circle1 subsisting in its true perfection1 more impartible than any centre1 and the triangle without interval2 and lastly1 by ascending into an union with herself1 of surveying every object which is subject to the power of cognition4 ,the figure1 therefore1 which is self-motive1 precedes that which is moved by another2 and the impartible that which is self-motive3 but that which is the same with one1 precedes the impartible itself4 ,for all things are bounded1 when they return to the unities of their nature2 since all things pass through these as a divine entrance into being4 ,and thus much for this long digression1 which we have delivered according to the sentiments of the ,pythagoreans4 ,but the geometrician1 contemplating that figure which is seated in the phantasy1 and defining this1 in the first place1 7since this definition agrees with sensibles1 in the second place7 says1 that figure is that which is comprehended by one or more boundaries4 ,for1 since he receives it together with matter1 and conceives of it as distant with intervals1 he does not improperly call it finite and terminated#afd4 ,since every thing which contains either intelligible or feasible matter1 is allotted an adventitious bound2 and is not itself bound1 but that which is bounded4 ,nor is it the bound of itself2 but one of its powers is terminating1 and the other terminated4 ,nor does it subsist in bound itself1 but is contained by bound4 ,for figure is joined to quantity1 and subsists together with it2 and1 at the same time1 quantity is subjected to figure2 but the reason and aspect of that quantity is nothing else than figure and form4 ,since1 indeed1 reason terminates quantity1 and adds to it a particular character and bound1 either simple or composite4 ,for1 since this also exhibits the twofold progression of bound and infinite in its proper forms1 7in the same manner as the reason of an angle17 it invests the objects of its comprehension with one boundary and simple form1 according to bound1 but with many1 according to infinity#afe4 ,hence1 every thing figured1 vindicates to itself either one boundary1 or a many4 ,euclid1 therefore1 denominating that which is figured and material1 and annexed to quantity figure1 does not improperly say1 that it is contained by one or more terms4 ,but ,possidonius defines figure to be concluding bound1 separating the reason of figure from quantity2 and considering it as the cause of terminating1 defining1 and comprehending quantity4 ,for that which encloses1 is different from that which is enclosed2 and bound from that which is bounded4 ,and ,possidonius1 indeed1 seems to regard the external surrounding bound2 but ,euclid1 the whole subject4 ,hence1 the one calls a circle a figure1 with relation to its whole plane1 and exterior ambit2 but the other with relation to its circumference only4 ,and the one defines that which is figured1 and which is beheld together with its subject3 but the other desires to define the reason of the circle2 ,i mean that which terminates and concludes its quantity4 ,but if any logician1 and captious person1 should blame the definition of ,euclid1 because he defines genus from species 7for things contained by one or more terms1 are the species of figure17 we shall assert1 in opposition to such an objection1 that genera also pre-occupy in themselves the powers of species4 ,and when men of ancient authority1 were willing to manifest genera themselves1 from those powers which genera contain1 they appeared1 indeed1 to enter on their design from species1 but1 in reality1 they explained genera from themselves1 and from the powers which they contain4 ,the reason of figure1 therefore1 since it is one1 comprehends the differences of many figures1 according to the bound and infinity residing in its nature4 ,and he who defined this reason1 was not void of understanding1 whilst he comprehended in a definition1 the differences of the powers it contained4 ,but you will ask1 ,from whence does the reason of figure originate1 and by what causes is it perfected8 ,i answer1 that it first arises from bound and infinite1 and that which is mixed from these4 ,hence it produces some species from bound1 others from infinite1 and others from the mixt4 ,and this it accomplishes by bringing the form of bound to circles2 but that of infinite1 to right lines3 and that of the mixt to figures composed from right and circular lines4 ,but1 in the second place1 this reason is perfected from that totality1 which is separated into dissimilar parts4 ,from whence1 indeed1 it occasions a whole to every form1 and each figure is cut into different species4 ,for a circle1 and every right-lined figure may be divided1 by reason or proportion1 into dissimilar figures2 which is the business of ,euclid in his book of divisions1 where he divides one figure into figures similar to each as are given2 but another into such as are dissimilar4 ,in the third place1 it is invigorated from accumulated multitude1 and1 on account of this1 extends forms of every kind1 and produces the multiform reasons of figures4 ,hence1 in propagating itself1 it does not cease till it arrives at something last1 and has unfolded all the variety of forms4 ,and1 as in the intelligible world1 one is shewn to abide in that which is2 and1 at the same time1 that which is in one1 so likewise1 reason exhibits circular in right-lined figures2 and on the contrary1 rectilinear comprehended in circular figures4 ,and it peculiarly manifests its whole nature in each1 and all these in all4 ,since the whole subsists in all collectively1 and in each separate and apart4 ,from that order1 therefore1 it is endued with this power4 ,in the fourth place1 it receives from the first of numbers#aff1 the measures of the progression of forms4 ,from whence it constitutes all figures according to numbers2 some1 indeed1 according to the more simple1 but others according to the more composite4 ,for triangles1 quadrangles1 quinquangles1 and all multangles1 proceed in infinitum1 together with the mutations of numbers4 ,but the cause of this is1 indeed1 unknown to the vulgar1 though1 to those who understand where number and figure subsist1 the reason is manifest4 ,fifthly1 it is replete with that division of forms1 which divides forms into other similar forms1 from another second totality1 which is also distributed into similar parts4 ,and by this1 a triangular reason is divided into triangles1 and a quadrangular reason into quadrangles4 ,and hence1 exercising our inward powers1 we effect what ,i have said in images1 since it pre-existed by far the first in its principles4 ,but by regarding these distributions1 we may render many causes of figures1 reducing them to their first principles4 ,and the more common1 or geometrical figure1 is allotted an order of this kind1 and from so many causes1 receives the perfection of its nature4 ,but1 from hence it advances to the genera of the gods1 and is variously attributed according to its various forms1 and energizes differently in different gods4 ,to some1 indeed1 affording more simple figures2 but to others1 such as are more composite4 ,and to some1 again1 assigning primary figures1 and those which are produced in superficies2 but to others 7entering the tumor of solid bodies7 such figures1 as in solids are convenient to themselves4 ,for all figures1 indeed1 subsist in all1 since the forms of the gods are accumulated1 and full of universal powers3 but1 by their peculiarity1 they produce one thing according to another4 ,for one possesses all things circularly1 another in a triangular manner1 but another according to a quadrangular reason4 ,and in a similar manner in solids4 ,,definition ,,xv4 ,a ,,circle is a ,plane ,figure1 comprehended by one ,line1 which is called the ,circumference1 to which all ,right ,lines falling from a certain ,point within the ,figure1 are equal to each other4 ,,definition ,,xvi4 ,and that ,point is called the ,,centre of the ,,circle4 ,a ,circle is the first1 most simple1 and most perfect of figures4 ,for it excels all solids1 because it exists in a more simple place2 but it is superior to the figures subsisting in planes1 on account of its similitude and identity4 ,and it has a corresponding proportion to bound1 and unity1 and a better co-ordination of being4 ,hence1 in a distribution of mundane and super-mundane figures1 you will always find that the circle is of a diviner nature4 ,for if you make a division into the heavens1 and the universal regions of generation1 you must assign to the heavens a circular form2 but to generation1 that of a right line1 ,for whatever among generable natures is circular1 descends from the heavens2 since generation revolves into itself1 through their circumvolutions1 and reduces its unstable mutation to a regular and orderly continuance4 ,but if you distribute incorporeal natures into soul and intellect1 you will say1 that the circle belongs to intellect1 and the right line to the soul4 ,and on this account1 the soul1 by its conversion to intellect1 is said to be circularly moved2 and it possesses the same proportion to intellect1 as generation to the heavens4 ,for it is circularly moved1 7says ,socrates#afg17 because it imitates intellect4 ,but the generation and progression of soul is made according to a right-line4 ,for it is the property1 of the soul to apply herself at different times to different forms4 ,but if you wish to divide into body and soul1 you must constitute every thing corporeal1 according to the right line2 but you must assign to every animal a participation of the identity and similitude of the circle4 ,for body is a composite1 and is endued with various powers1 similar to right-lined figures3 but soul is simple and intelligent2 self-motive1 and self-operative2 converted into1 and energizing in herself4 ,from whence1 indeed1 ,timus also1 when he had composed the elements of the universe from right-lined figures1 assigned to them a circular motion and formation1 from that divine soul which is seated in the bosom of the world4 ,and thus1 that the circle every where holds the first rank1 in respect of other figures1 is sufficiently evident from the preceding observations4 ,but it is requisite to survey its whole series1 beginning supernally1 ending in inferiors1 and perfecting all things1 according to the aptitude of the natures which receive its alliance4 ,to the gods1 therefore1 it affords a conversion to their causes1 and ineffable union3 it occasions their abiding in themselves1 prevents their departing from their own beatitude1 strengthens their highest unions1 as centres desirable to inferior natures2 and stably places about these the multitude of the powers which the gods possess1 containing them in the simplicity of their essences4 ,but the circle affords to intellectual natures1 a perpetual energy in themselves1 is the cause of their being filled with knowledge from themselves1 and of possessing in their essences1 intelligibles contractedly2 and of perfecting intellections in themselves4 ,for every intellect1 proposes to itself that which is intelligible2 and this is as a centre to intellect1 about which it continually revolves3 for intellect folds itself1 and operates about this1 and is united within itself on all sides1 by universal intellectual energies4 ,but it extends to souls by illumination1 a self-vital1 and self-motive power1 and an ability of turning1 and leaping round intellect1 and of returning according to proper convolutions1 unfolding the impartibility of intellect4 ,again1 the intellectual orders excel souls after the manner of centres1 but souls energize circularly about their nature4 ,for every soul1 according to its intellectual part1 and the supreme one1 which is the very flower of its essence1 receives a centre3 but1 according to its multitude1 it has a circular revolution1 desiring1 by this means1 to embrace the intellect which it participates4 ,but1 to the celestial bodies1 the circle affords an assimilation to intellect1 equality1 a comprehension of the universe1 in proper limits1 revolutions which take place in determinate measures1 a perpetual subsistence1 a nature without beginning and end1 and every thing of this kind4 ,and to the elements under the concave of the moon's orb1 it is the cause of a period1 conversant with mutations2 an assimilation to the heavens2 that which is without generation1 in generated natures2 that which abides in things which are moved2 and whatever is bounded in partible essences4 ,for all things are perpetual1 through the circle of generation2 and equability is every where preserved on account of the reciprocation of corruption4 ,since1 if generation did not return1 in a circular revolution1 in a short space of time1 the order1 and all the ornament of the elements would vanish4 ,but again1 the circle procures to animals and plants1 that similitude which is found in generations2 for these are produced from seeds1 and seeds from these4 ,hence1 generation here1 and a circumvolution1 alternately takes place1 from the imperfect to the perfect1 and the contrary2 so that corruption subsists together with generation4 ,but1 besides this1 to unnatural productions it imposes order1 and reduces their indeterminate variety to the limitation of bound2 and1 through this1 nature herself is gracefully ornamented in the last vestiges of her powers4 ,hence1 things contrary to nature have a revolution according to determinate numbers1 and not only fertility1 but also sterility1 subsists according to the alternate convolutions of circles 7as the discourse of the ,muses evinces71 and all evils though they are dismissed from the presence of the gods1 into the place of mortals1 yet these roll round1 says ,socrates1 and to these there is present a circular revolution1 and a circular order2 so that nothing immoderate and evil is deserted by the gods2 but that providence1 which is perfective of the universe1 reduces also the infinite variety of evils1 to bound1 and an order convenient to their nature4 ,the circle1 therefore1 is the cause of ornament to all things1 even to the last participations1 and leaves nothing destitute of itself1 since it supplies beauty1 similitude1 formation1 and perfection to the universe4 ,hence too1 in numbers it contains the middle centres of the whole progression of numbers1 which revolves from unity to the decad 7or ten74 ,for five and six exhibit a circular power1 because1 in the progressions from themselves1 they return again into themselves1 as is evident in the multiplication of these numbers4 ,multiplication1 therefore1 is an image of progression1 since it is extended into multitude2 but an ending in the same species1 is an image of regression into themselves4 ,but a circular power affords each of these1 exciting1 indeed1 as from an abiding centre1 those causes which are productive of multitude2 but converting multitude after the productions to their causes4 ,two numbers1 therefore1 having the properties of a circle1 possess the middle place between all numbers3 of which one1 indeed1 precedes every convertible genus of males and an odd nature2 but the other1 recalls every thing feminine and even1 and all prolific series1 to their proper principles1 according to a circular power4 ,and thus much concerning the perfection of the circle4 ,let us now contemplate the mathematical definition of the circle1 which is every way perfect4 ,in the first place1 therefore1 he defines it a figure1 because1 indeed1 it is finite1 and every where comprehended by one limit1 and is not of an infinite nature1 but associated to bound4 ,likewise plane1 because1 since figures are either beheld in superficies1 or in solid bodies1 a circle is the first of plane figures1 excelling solids in simplicity1 but possessing the proportion of unity to planes4 ,but comprehended by one line1 because it is similar to one1 by which it is defined1 and because it does not extrinsically receive a variety of surrounding terms4 ,and again1 that this line makes all the lines drawn to it from a certain point within equal1 because of the figures which are bounded by one line1 some have all the lines proceeding from the middle equal2 but others not at all4 ,for the ellipsis is comprehended by one line1 yet all the lines issuing from the centre1 and bounded by is curvature1 are not equal1 but only two4 ,also the plane1 which is included by the line called a cissoid1 has one containing line1 yet it does not contain a centre1 from which all the lines are equal4 ,but1 because the centre in a circle is entirely one point 7for there are not many centres of one circle71 on this account1 the geometrician adds1 that lines falling from one point to the bound of the circle1 are equal4 ,for there are infinite points within it1 but of all these1 one only has the power of a centre4 ,and because this one point1 from which all the lines drawn to the circumference of the circle are equal1 is either within the circle1 or without 7for every circle has a pole1 from which all the lines drawn to its circumference are equal71 on this account he adds1 of the points within the figure1 because1 here he receives the centre alone1 and not the pole4 ,for he wishes to behold all its properties in one plane1 but the pole is more elevated than the subject plane4 ,hence1 he necessarily adds1 in the end of the definition1 that this point1 which is placed within the circle1 and to which all right lines drawn from it to the circumference1 are equal1 is the centre of the circle4 ,for there are only two points of this kind1 the pole and the centre4 ,but the former is without1 and the other within the plane4 ,thus1 for instance1 if you conceive a perpendicular standing on the centre of a circle1 its superior extremity is the pole3 for all lines drawn from it to the circumference of the circle1 are demonstrated to be equal4 ,and1 in like manner1 in a cone1 the vertex of the whole cone1 is the pole of the circle at the base4 ,and thus far we have determined what a circle is1 and its centre1 and what the nature is of its circumference1 and the whole circular figure4 ,again1 therefore1 from these1 let us return to the speculation of their exemplars1 contemplating in them the centre1 according to one impartible and stable excellence4 ,but the distances from the centre1 according to the progressions which are made from one1 to multitude infinite in capacity4 ,and the circumference of the circle1 according to the regression of the progressions to the centre1 by means of which the multitude of powers are rolled round their union1 and all of them hasten to its comprehension1 and desire to energize about its indivisible embrace4 ,and1 as in the circle itself1 all things subsist together1 the centre1 intervals1 and external circumference2 so in these which are its image1 one thing has not an essence pre-existent1 and another consequent in time2 but all things are1 indeed1 together1 permanency1 progression1 and regression4 ,but these differ from those1 because the former subsist indivisibly1 and without any dimension2 but the latter with dimension2 and in a divisible manner2 the centre existing in one place1 the lines emanating from the centre1 in another2 and the external circumference terminating the circle1 having a still different situation4 ,but there all things abide in one3 for if you regard that which performs the office of a centre1 you will find it the receptacle of all things4 ,if the progression distant from the centre1 in this1 likewise1 you will find all things contained4 ,and1 in a similar manner1 if you regard its regression4 ,when1 therefore1 you are able to perceive all things subsisting together1 and have taken away the defect proceeding from dimension1 and have removed from your inward vision1 the position about which partition subsists1 you will find the true circle1 advancing to itself1 bounding1 and energizing in itself1 existing both one and many1 and abiding1 proceeding1 and returning2 likewise firmly establishing that part of its essence which is most impartible1 and especially singular2 but advancing from this according to rectitude1 and the infinity which it contains2 and rolling itself from itself to one1 and exciting itself by similitude and identity to the impartible centre of its nature1 and to the occult power of the one which it contains4 ,but this one1 which the circle contains1 and environs in its bosom1 it emulates according to the multitude of its own nature4 ,for that which is convolved1 imitates that which abides1 and the periphery is as a centre which is distant with interval1 and nods to itself1 hastening to receive1 and to become one with the centre1 and to terminate its regress where it received the principle of its progression4 ,for the centre is every where in the place of that which is lovely1 and the object of desire1 presiding over all things which subsist about its nature1 and existing as the beginning and author of all progressions4 ,and this the mathematical centre also expresses1 by terminating all the lines falling from itself to the circumference1 and by affording to them equality1 as an image of proper union4 ,but the oracles likewise define the centre1 after this manner3 ,the centre is that from which and#afh to which all the lines to the circumference are equal4 ,indicating the beginning of the distance of the lines1 by the particle from which2 but the middle of the circumference by the particle to which3 for this1 in every part1 is joined with the centre4 ,but if it be necessary to declare the first cause1 through which a circular figure appears and receives its perfection1 ,i affirm1 that it is the supreme order of intelligibles4 ,for the centre1 indeed1 is assimilated to the cause of bound2 but the lines emanating from this1 and which are infinite1 with respect to themselves1 both in multitude and magnitude1 represent infinity2 and the line which terminates their extension1 and conjoins the circular figure with the centre1 is similar to that occult ornament1 consisting from the intelligible orders2 which ,orpheus also says1 is circularly borne1 in the following words1 ,but it is carried with an unwearied energy1 according to an infinite circle4 ,for1 since it is moved intelligibly1 about that which is intelligible1 having it for the centre of its motion1 it is1 with great propriety1 said to energize in a circular manner4 ,hence1 from these also1 the triadic god#afi proceeds1 who contains in himself the cause of the progression of right-lined figures4 ,for on this account1 wise men1 and the most mystic of theologists1 have fabricated his name4 #agj;,hence too1 it is manifest1 that a circle is the first of all figures3 but a triangle is the first of such as are right-lined4 ,figures1 therefore1 appear first in the regular ornaments of the gods2 but they have a latent subsistence1 according to pre-existent causes1 in intelligible essences4 ,,definition ,,xvii4 ,a ,,diameter of a ,circle is a certain straight ,line1 drawn through the ,centre1 which is terminated both ways by the ,circumference of the ,circle1 and1 divides the ,circle into two equal ,parts4 ,euclid here perspicuously shews1 that he does not define every diameter1 but that which belongs to a circle only4 ,because there is a diameter of quadrangles and all parallelograms1 and likewise of a sphere among solid figures4 ,but in the first of these1 it is denominated a diagonal3 but in a sphere1 the axis2 and in circles the diameter only4 ,indeed1 we are accustomed to speak of the axis of an ellipsis1 cylinder1 and cone2 but of a circle1 with propriety1 the diameter4 ,this1 therefore1 in its genus1 is a right-line2 but as there are many right-lines in a circle1 as likewise infinite points1 one of which is a centre1 so this only is called a diameter1 which passes through the centre1 and neither falls within the circumference1 nor transcends its boundary2 but is both ways terminated by its comprehensive bound4 ,and these observations exhibit its origin4 ,but that which is added in the end1 that it also divides the circle into two equal parts1 indicates its proper energy in the circle1 exclusive of all other lines drawn through the centre1 which are not terminated both ways by the circumference4 ,but they report1 that ,thales first demonstrated1 that the circle was bisected by the diameter4 ,and the cause of this bisection1 is the indeclineable transit of the right line1 through the centre4 ,for1 since it is drawn through the middle1 and always preserves the same inflexible motion1 according to all its parts1 it cuts off equal portions on both sides to the circumference of the circle4 ,but if you desire to exhibit the same mathematically1 conceive the diameter drawn1 and one part of the circle placed on the other#aga4 ,then1 if it is not equal1 it either falls within1 or without2 but the consequence either of these ways must be1 that a less right-line will be equal to a greater4 ,since all lines from the centre to the circumference are equal4 ,the line1 therefore1 which tends to the exterior circumference1 will be equal to that which tends to the interior4 ,but this is impossible4 ,these parts of the circle1 then1 agree1 and are on this account equal4 ,but here a doubt arises1 if two semi-circles are produced by one diameter1 and infinite diameters may be drawn through the centre1 a double of infinities will take place1 according to number4 ,for this is objected#agb by some against the section of magnitudes to infinity4 ,but this we may solve by affirming1 that magnitude may1 indeed1 be divided infinitely1 but not into infinites4 ,for this latter mode produces infinites in energy1 but the former in capacity only4 ,and the one affords essence to infinite1 but the other is the source of its origin alone4 ,two semi-circles1 therefore1 subsist together with one diameter1 yet there will never be infinite diameters1 although they may be infinitely assumed4 ,hence1 there can never be doubles of infinites2 but the doubles which are continually produced1 are the doubles of finites2 for the diameters which are always assumed1 are finite in number4 ,and what reason can be assigned why every magnitude should not have finite divisions1 since number is prior to magnitudes1 defines all their sections1 pre-occupies infinity1 and always determines the parts which rise into energy1 from dormant capacity8 ,,definition ,,xviii4 ,a ,,semi-,,circle is the ,figure contained by the ,diameter1 and that ,part of the ,circumference which is cut off by the ,diameter4 ,,definition ,,xix4#agc ,but the ,,centre of the ,semi-circle1 is the same with that of the ,circle4 ,from the definition of a circle ,euclid finds out the nature of the centre1 differing from all the other points which the circle contains4 ,but from the centre he defines the diameter1 and separates it from the other right lines1 which are described within the circle4 ,and from the diameter1 he teaches the nature of the semi-circle2 and informs us1 that it is contained by two terms1 always differing from each other1 viz4 a right-line and a circumference3 and that this right-line is not any one indifferently1 but the diameter of the circle4 ,for both a less and a greater segment of a circle1 are contained by a right-line and circumference2 yet these are not semi-circles1 because the division of the circle is not made through the centre4 ,all these figures1 therefore1 are biformed1 as a circle was monadic1 and are composed from dissimilars4 ,for every figure which is comprehended by two terms1 is either contained by two circumferences1 as the lunular3 or by a right-line and circumference1 as the above mentioned figures2 or by two mixt lines1 as if two ellipses intersect each other 7since they enclose a figure1 which is intercepted between them71 or by a mixt line and circumference1 as when a circle cuts an ellipsis2 or by a mixt and right-line1 as the half of an ellipsis4 ,but a semi-circle is composed from dissimilar lines1 yet such as are1 at the same time1 simple1 and touching each other by apposition4 ,hence1 before he defines triadic figures1 he1 with great propriety1 passes from the circle to a biformed figure4 ,for two right-lines can1 indeed1 never comprehend space4 ,but this may be effected by a right-line and circumference4 ,likewise by two circumferences1 either making angles1 as in the lunular figure2 or forming a figure without angles1 as that which is comprehended by concentric circles4 ,for the middle space intercepted between both1 is comprehended by two circumferences2 one interior1 but the other exterior1 and no angle is produced4 ,for they do not mutually intersect1 as in the lunular figure1 and that which is on both sides convex4 ,but that the centre of the semi-circle is the same with that of the circle1 is manifest4 ,for the diameter1 containing in itself the centre1 completes the semi-circle1 and from this all lines drawn to the semi-circumference are equal4 ,for this is a part of the circumference of the circle4 ,but equal right lines proceed from the centre to all parts of the circumference4 ,the centre1 therefore1 of the circle and semi-circle is one and the same4 ,and it must be observed1 that among all figures1 this alone contains the centre in its own perimeter1 ,i say1 among all plane figures4 ,hence you may collect1 that the centre has three places4 ,for it is either within a figure1 as in the circle2 or in its perimeter1 as in the semi-circle2 or without the figure1 as in certain conic lines#agd4 ,what then is indicated by the semi-circles1 having the same centre with the circle1 or of what things does it bear an image1 unless that all figures which do not entirely depart from such as are first1 but participate them after a manner1 may be concentric with them1 and participate of the same causes8 ,for the semi-circle communicates with the circle doubly1 as well according to the diameter1 as according to the circumference4 ,on this account1 they possess a centre also in common4 ,and perhaps1 after the most simple principles1 the semi-circle is assimilated to the second co-ordinations1 which participate those principles2 and by their relation to them1 although imperfectly1 and by halves1 they are1 nevertheless1 reduced to that which is1 and to their first original cause4 ,,definition ,,xx4 ,,rectilinear ,,figures are those which are comprehended by ,straight ,lines4 ,,definition ,,xxi4 ,,trilateral ,,figures1 or ,,triangles1 by three ,straight ,lines4 ,,definition ,,xxii4 ,,quadrilateral1 by four ,straight ,lines4 ,,definition ,,xxiii4 ,,multilateral ,,figures1 or ,,polygons1 by more than four ,straight ,lines4 ,after the monadic figure having the relation of a principle to all figures1 and the biformed semi-circle1 the progression of right-lined figures in infinitum1 according to numbers1 is delivered4 ,for on this account also1 mention was made of the semi-circle1 as communicating according to terms or boundaries2 partly1 indeed1 with the circle1 but partly with right-lines3 just as the duad is the medium between unity and number4 ,for unity1 by composition1 produces more than by multiplication2 but number1 on the contrary1 is more increased by multiplication than composition3 and the duad1 whether multiplied into1 or compounded with itself1 produces an equal quantity4 ,as1 therefore1 the duad is the middle of unity and number1 so likewise1 a semi-circle communicates1 according to its base1 with right-lines2 but according to its circumference1 with the circle4 ,but right-lined figures proceed orderly to infinity1 attended by number and its bounding power1 which begins from the triad4 ,on this account1 ,euclid also begins from hence#age4 ,for he says1 trilateral and quadrilateral1 and the following figures1 called by the common name of multilateral3 since trilateral figures are also multilaterals2 but they have likewise a proper1 besides a common denomination4 ,but1 as we are but little able to pursue the rest1 on account of the infinite progression of numbers1 we must be content with a common denomination4 ,but he only makes mention of trilaterals and quadrilaterals1 because the triad and tetrad are the first in the order of numbers2 the former being a pure odd among the odd2 but the latter1 an entire even among even numbers4 ,euclid1 therefore1 assumes both in the origin of right-lined figures1 for the purpose of exhibiting their subsistence1 according to all even and odd numbers4 ,besides1 since he is about to teach concerning these in the first book1 as especially elementary 7,i mean triangles and parallelograms7 he does not undeservedly1 as far as to these1 establish a proper enumeration3 but he embraces all other right-lined figures by a common name1 calling them multilaterals3 but of these enough4 ,again1 assuming a more elevated exordium1 we must say1 that of plane figures1 some are contained by simple lines1 others by such as are mixt1 but others again by both4 ,and of those which are comprehended by simple lines1 some are contained by similars in species1 as right-lines2 but others by dissimilars in species1 as semi-circles1 and segments1 and apsides1 which are less than semi-circles4 ,likewise of those which are contained by similars in species1 some are comprehended by a circular line2 but others by a right-line4 ,and of those comprehended by a circular line1 some are contained by one1 others by two1 but others by more than two4 ,by one1 indeed1 the circle itself4 ,but by two1 some without angles1 as the crowns#agf terminated by concentric circles2 but others angular 77 as the lunula4 ,and of those comprehended by more than two1 there is an infinite procession4 ,for there are certain figures contained by three and four and succeeding circumferences4 ,thus1 if three circles touch each other1 they will intercept a certain trilateral space2 but if four1 one terminated by four circumferences1 and in like manner1 by a successive progression4 ,but of those contained by right lines1 some are comprehended by three1 others by four1 and others by a multitude of lines4 ,for neither is space comprehended by two right-lines1 nor much more by one right-line4 ,hence1 every space comprehended by one boundary1 or by two1 is either mixt or circular4 ,and it is mixt in a twofold manner1 either because the mixt lines comprehend it1 as the space intercepted by the cissoidal line2 or because it is contained by lines dissimilar in species1 as the apsis3 since mingling is twofold1 either by apposition or confusion4 ,every right-lined figure1 therefore1 is either trilateral1 or quadrilateral1 or gradually multilateral2 but every trilateral1 or quadrilateral1 or multilateral figure1 is not right-lined2 since so great a number of sides is also produced from circumferences4 ,and thus much concerning the division of plane figures4 ,but we have already asserted#agg1 that rectitude of progression is both a symbol of motion and infinity1 and that it is peculiar to the generative co-ordinations of the gods1 and to the producers of difference1 and to the authors of mutation and motion4 ,right-lined figures1 therefore1 are peculiar to these gods1 who are the principles of the prolific energy of the whole progression of forms4 ,on which account1 generation also1 was principally adorned by these figures1 and is allotted its essence from these1 so far as it subsists in continual motion and mutation without end4 ,,definition ,,xxiv4 ,of three-sided ,,figures3 an ,,equilateral ,,triangle is that which has three equal ,sides4 ,,definition ,,xxv4 ,an ,,isosceles ,,triangle1 is that which has only two ,sides equal4 ,,definition ,,xxvi4 ,a ,,scalene ,,triangle1 is that which has three unequal ,sides4 ,,definition ,,xxvii4 ,a ,,right-,,angled ,,triangle is that which has a ,right ,angle4 ,,definition ,,xxviii4 ,an ,,obtuse-,,angled ,,triangle is that which has an ,obtuse ,angle4 ,,definition ,,xxix4 ,an ,,acute-,,angled ,,triangle is that which has three ,acute ,angles4 ,the division of triangles sometimes commences from angles1 but sometimes from sides4 ,and that1 indeed1 which originates from sides1 precedes as known2 but that from angles follows as a proper distribution4 ,for these three angles alone belong to right-lined figures1 viz4 the right1 the obtuse1 and the acute3 but the equality and inequality of sides subsist also in non-rectilinear figures4 ,euclid says1 therefore1 that of triangles1 some are equilateral1 others isosceles1 and others scalene3 for they have either all their sides equal1 or all unequal1 or only two equal4 ,and again1 that of triangles some are right-angled1 others obtuse-angled1 and others acute-angled4 ,and he defines a right-angled triangle1 that which has one right angle1 as likewise an obtuse-angled triangle1 that which has one obtuse angle3 for it is impossible that a triangle can have more than one right1 or obtuse angle#agh4 ,but he defines an acute-angled triangle1 that which has all its angles acute4 ,for here it is not sufficient that it should have only one acute2 since1 in this case1 all triangles would be acute-angled1 as every triangle has necessarily two acute angles#agi4 ,but1 to possess three acute angles1 is the property of an acute-angled triangle alone4 ,but ,euclid appears to me to have made a separate division into angles and sides1 from considering this alone1 that every triangle is not also trilateral4 ,for there are quadrilateral triangles1 which are called by mathematicians themselves 77 that is1 similar to the point of a spear#ahj3 but by ,zenodorus 77 that is1 having an hollow angle4 ,for on one of the sides of a trilateral figure1 constitute two right-lines inwardly2 by this means a certain space will be enclosed1 which is comprehended by external and internal right-lines1 and which has three angles2 one1 indeed1 contained by the external lines2 but two comprehended by these and the internal lines1 at the extremities in which these lines are conjoined4 ,a figure of this kind1 therefore1 is a quadrilateral triangle4 ,and hence1 it does not immediately follow1 that because a figure has three angles 7whether they are all acute1 or one right1 or one obtuse71 we shall find it trilateral2 for it may be1 perhaps1 quadrilateral4 ,in like manner1 you may also find quadrangles having more than four sides4 ,and therefore1 we must not rashly determine the number of sides from the multitude of angles4 ,but of this enough4 ,but the ,pythagoreans affirm that the triangle is simply the principle of generation1 and of the formation of generable natures4 ,on which account1 ,timus says1 that natural reasons1 as well as those of the construction of the elements1 are triangular4 ,for they are distant by a triple interval1 are on all sides collective of partible1 and variously mutable natures1 are replete with material infinity1 and bear before themselves the conjunctions of material bodies1 loosened and free3 as1 indeed1 triangles also are comprehended by three right-lines1 but they possess angles which collect the multitude of lines1 and afford to them an adventitious angle and conjunction4 ,with great propriety1 therefore1 ,philolaus has consecrated the angle of a triangle to four gods1 ,saturn1 ,pluto1 ,mars1 and ,bacchus1 comprehending in these the whole quadripartite ornament of the elements descending from the heavens1 or from the four segments of the zodiac4 ,for ,saturn constitutes an essence wholly humid and frigid2 but ,mars a nature totally fiery2 and ,pluto contains the whole terrestrial life2 but ,bacchus governs a humid and hot generation2 of which wine also is a symbol1 for this is humid and hot4 ,hence1 all these gods differ according to their operations in inferior concerns3 but they are mutually united according to their proper natures4 ,and on this account1 ,philolaus collects their union according to one angle4 ,but if the differences of triangles contribute to generation1 we shall very properly confess that a triangle is the principle and author of the constitution of sublunary natures4 ,for a right angle1 indeed1 affords them essence1 and determines the measure of being2 and the reason of a right-angled triangle produces the essence of the elements of generable natures2 but an obtuse angle assigns to them universal distance2 and the reason of an obtuse-angled triangle increases material forms in magnitude1 and in mutation of every kind4 ,but an acute angle effects their divisible nature2 and the reason of an acute-angled triangle prepares them to receive infinite division4 ,but simply1 a triangular reason constitutes the essence of material bodies distant with interval1 and on all sides divisible4 ,and thus much should we speculate concerning the nature of triangles4 ,but from these divisions you may understand1 that all the species of triangles are neither more nor less than seven4 ,for the equilateral triangle is one1 since it is acute-angled only2 but each of the rest is triple4 ,for the isosceles is either right-angled1 or obtuse-angled1 or acute-angled2 and1 in like manner1 the scalene triangle possesses this triple difference4 ,if then1 these have a triple distinction1 but the equilateral has but one mode of existence1 all the species of triangles will be seven4 ,but again1 you will understand the proportion of triangles to the things which are1 according to the division of sides2 for the equilateral1 entirely excelling in equality and simplicity1 is allied to divine souls2 since it is the measure and equality of things unequal1 in the same manner as divinity of all inferior concerns4 ,but the isosceles triangle is allied to the better genera1 which govern a material nature1 the greater part of which genera is held by the limitation of measure2 but their extremes extend to inequality and material immoderation2 for the two sides of an isosceles triangle are equal1 but the base is unequal4 ,but a scalene triangle symbolizes with partible lives1 which are on all sides lame and defective1 which prepare themselves for generation1 and are replete with matter and material imperfection4 ,,definition ,,xxx4 ,of ,quadrilateral ,figures1 a ,,quadrangle or ,,square is that which has all its ,sides equal1 and all its ,angles ,right ,angles4 ,,definition ,,xxxi4 ,an ,,oblong is that which has all its ,angles right ,angles1 but has not all its ,sides equal4 ,,definition ,,xxxii4 ,a ,,rhombus1 is that which has all its ,sides equal1 but its ,angles are not right ,angles4 ,,definition ,,xxxiii4 ,a ,,rhomboid is that which has its opposite ,sides equal to one another1 but all its ,sides are not equal1 nor its ,angles ,right ,angles4 ,,definition ,,xxxiv4 ,all other ,quadrilateral ,figures besides these1 are called ,,trapeziums4 ,it is requisite that the first division of quadrilateral figures should take place in two numbers2 and that some of them should be called parallelograms1 but others non-parallelograms4 ,but of parallelograms some are rectangular and equilateral1 as quadrangles2 but others neither of these1 as rhomboids3 others again1 are rectangular1 but not equilateral1 as oblongs3 but others1 on the contrary1 are equilateral1 but not rectangular1 as the rhombuses4 ,for it is requisite either to possess both1 viz4 equality of sides and rectitude of angles1 or neither2 or one of these1 and this in a twofold respect4 ,hence a parallelogram has a quadruple subsistence4 ,but of non-parallelograms1 some have only two parallel sides1 and not the rest2 but others have none of their sides parallel4 ,and those are called ,trapeziums1 but these ,trapezoids4 ,but of ,trapeziums1 some1 indeed1 have the sides equal1 by which the parallel sides of this kind are conjoined2 but others unequal2 and the former of these are called isosceles trapeziums2 but the latter scalene trapeziums4 ,a quadrilateral figure1 therefore1 is constituted by us according to a seven-fold distribution4 ,for one is a quadrangle2 but the other an oblong2 the third a rhombus2 the fourth a rhomboides2 the fifth an isosceles trapezium2 the sixth a scalene trapezium2 the seventh a trapezoid4 ,but ,possidonius makes a perfect division of right-lined quadrilateral figures into so many members2 for he establishes seven species of these2 as likewise of triangles4 ,but ,euclid could not divide into parallelograms and non parallelograms1 because he neither mentions parallels1 nor teaches us concerning the parallelogram itself4 ,but trapeziums1 and all trapezoids1 he calls by a common name1 describing trapeziums themselves1 according to the difference of those four figures#aha1 in which the property of parallelograms is verified4 ,and this is to have the opposite sides and angles equal4 ,for a quadrangle and an oblong1 and a rhombus1 have their opposite sides and angles equal4 ,but in a rhomboides he only adds this1 that its opposite sides are equal1 lest he should define it by negations alone1 since he neither calls it equilateral1 nor rectangular4 ,for where we want proper appellations1 it is necessary to use such as are common4 ,but we should hear ,euclid shewing that this is common to all parallelograms4 ,but a rhombus appears to be a quadrangle having its sides moved1 and a rhomboides a moved oblong4 ,hence1 according to sides1 these do not differ from those2 but they vary only according to the obtuseness and acuteness of angles2 since the quadrangle and the oblong are rectangular4 ,for if you conceive a quadrangle or an oblong1 having its sides drawn in such a manner1 that while two of its opposite angles are dilated1 the other two are contracted2 then the dilated angles will appear obtuse1 and the contracted1 acute4 ,and the appellation of rhombus#ahb seems to have been imposed from motion4 ,for if you conceive a quadrangle moving after the manner of a rhombus1 it will appear to you changed in order1 according to its angles3 just as if a circle is moved after the manner of a sling1 it will immediately exhibit the appearance of an ellipsis4 ,but here you may perhaps enquire concerning the quadrangle1 why it has this denomination8 and why the appellation of quadrangle may not be applied to other quadrilateral figures1 as the name of triangle is common to all those which are neither equiangular nor equilateral1 and in like manner of quinquangles or pentagons2 for the geometrician1 in these1 adds only the particle an equilateral triangle1 or a quinquangle1 which is equilateral and equiangular1 as if these could not be otherwise than such as they are8 ,but when he mentions a quadrangle1 he immediately indicates that it must be equilateral and rectangular4 ,but the reason of this is as follows3 a quadrangle alone has the best space1 both according to its sides and angles4 ,for each of the latter is right1 intercepting a measure of angles1 which neither receives intention nor remission4 ,as it excels1 therefore1 in both respects1 it deservedly obtains a common appellation4 ,but a triangle1 though it may have equal sides1 yet will in this case have all its angles acute1 and a quinquangle all its angles obtuse4 ,since1 therefore1 of all quadrilateral figures1 a quadrangle alone is replete with equality of sides1 and rectitude of angles1 it was not undeservedly allotted this appellation3 for1 to excellent forms1 we often dedicate the name of the whole4 ,but it appeared also to the ,pythagoreans1 that this property of quadrilateral figures1 principally conveyed an image of a divine essence4 ,for they particularly signified by this1 a pure and immaculate order4 ,since rectitude imitates inflexibility1 but equality a firm and permanent power3 for motion emanates from inequality1 but quiet from equality itself4 ,the gods1 therefore1 who are the authors to all things of stable disposition1 of pure and uncontaminated order1 and of indeclinable power1 are deservedly manifested as from an image1 by a quadrangular figure4 ,but1 besides these1 ,philolaus also1 according to another apprehension1 calls a quadrangular angle1 the angle of ,rhea1 ,ceres and ,vesta4 ,for1 since a quadrangle constitutes the earth1 and is its proximate element1 as we learn from ,timus1 but the earth herself receives from all these divinities1 genital seeds1 and prolific powers1 he does not unjustly consecrate the angle of a quadrangle to these goddesses1 the bestowers of life4 ,for some call both the earth and ,ceres1 ,vesta#ahc1 and they say that ,rhea totally participates her nature1 and that all generative causes are contained in her essence4 ,philolaus1 therefore1 says that a quadrangular angle comprehends1 by a certain terrestrial power1 one union of the divine genera4 ,but some assimilate a quadrangle to universal virtue1 so far as every quadrangle from its perfection has four right angles4 ,just as we say that each of the virtues is perfect1 content with itself1 the measure and bound of life1 and the middle of every thing which1 in morals1 corresponds to the obtuse and acute4 ,but it is by no means proper to conceal1 that ,philolaus attributes a triangular angle to four1 but a quadrangular angle to three gods1 exhibiting their alternate transition1 and the community of all things in all1 of odd natures in the even1 and of even in the odd4 ,hence1 the tetradic ternary1 and the triadic quaternary1 participating of prolific and efficacious goods1 contain the whole ornament of generable natures1 and preserve them in their proper state4 ,from which the duodenary1 or the number twelve1 is excited to a singular unity1 viz4 the government of ,jupiter4 ,for ,philolaus says1 that the angle of a dodecagon 7or twelve-sided figure7 belongs to ,jove1 so far as ,jupiter contains and preserves1 by his singular union1 the whole number of the duodenary4 ,for also1 according to ,plato1 ,jupiter presides over the duodenary#ahd1 and governs and moderates the universe with absolute sway4 ,and thus much we have thought proper to discourse concerning quadrilateral figures1 as well declaring the sense of our author1 as likewise affording an occasion of more profound inspections to such as desire the knowledge of intelligible and occult essences4 ,,definition ,,xxxv4 ,,parallel ,,right ,,lines are such as being in the same ,plane1 and produced both ways infinitely1 will in no part mutually coincide4 ,what the elements of parallels are1 and by what accidents in these they may be known1 we shall afterwards learn3 but what parallel right lines are1 he defines in these words3 8,it is requisite1 therefore 7says he71 that they should be in one plane1 and while they are produced both ways have no coincidence1 but be extended in infinitum40 ,for non-parallel lines also1 if they are produced to a certain distance1 will not coincide4 ,but to be produced infinitely1 without coincidence1 expresses the property of parallels4 ,nor yet this absolutely1 but to be extended both ways infinitely1 and not coincide4 ,for it is possible that non-parallel lines may also be produced one way infinitely1 but not the other2 since1 verging in this part1 they are far distant from mutual coincidence in the other4 ,but the reason of this is1 because two right-lines cannot comprehend space2 for if they verge to each other both ways1 this cannot happen4 ,besides this1 he very properly considers the right-lines as subsisting in the same plane4 ,for if the one should be in a subject plane1 but the other in one elevated1 they will not mutually coincide according to every position1 yet they are not on this account parallel4 ,the plane1 therefore1 should be one1 and they should be produced both ways infinitely1 and not coincide in either part4 ,for with these conditions1 the right-lines will be parallel4 ,and agreeable to this1 ,euclid defines parallel right-lines4 ,but ,posidonius says1 parallel lines are such as neither incline nor diverge in one plane2 but have all the perpendiculars equal which are drawn from the points of the one to the other4 ,but such lines as make their perpendiculars always greater and less1 will some time or other coincide1 because they mutually verge to each other4 ,for a perpendicular is capable of bounding the altitudes of spaces1 and the distances of lines4 ,on which account1 when the perpendiculars are equal1 the distances of the right lines are also equal2 but when they are greater and less1 the distance also becomes greater and less1 and they mutually verge in those parts1 in which the lesser perpendiculars are found4 ,but it is requisite to know1 that non-coincidence does not entirely form parallel lines4 ,for the circumferences of concentric circles do not coincide3 but it is likewise requisite that they should be infinitely produced4 ,but this property is not only inherent in right1 but also in other lines3 for it is possible to conceive spirals described in order about right lines1 which if produced infinitely together with the right lines1 will never coincide#ahe4 ,geminus1 therefore1 makes a very proper division in this place1 affirming from the beginning1 that of lines some are bounded1 and contain figure1 as the circle and ellipsis1 likewise the cissoid1 and many others2 but others are indeterminate1 which may be produced infinitely1 as the right-line1 and the section of a right-angled1 and obtuse angled cone2 likewise the conchoid itself4 ,but again1 of those which may be produced in infinitum1 some comprehend no figure1 as the right-line and the conic sections2 but others1 returning into themselves1 and forming figure1 may afterwards be infinitely produced4 ,and of these some will not hereafter coincide1 which resist coincidence1 how far soever they may be produced2 but others are coincident1 which will some time or other coincide4 ,but of non-coincident lines1 some are mutually in one plane2 and others not4 ,and of non-coincidents subsisting in one plane1 some are always mutually distant by an equal interval2 but others always diminish the interval1 as an hyperbola in its inclination to a right-line1 and likewise the conchoid#ahf4 ,for these1 though they always diminish the interval1 never coincide4 ,and they mutually converge1 indeed1 but never perfectly nod to each other2 which is indeed a theorem in geometry especially admirable1 exhibiting certain lines endued with a non-assenting nod4 ,but the right-lines1 which are always distant by an equal interval1 and which never diminish the space placed between them in one plane1 are parallel lines4 ,and thus much we have extracted from the studies of the elegant ,geminus1 for the purpose of explaining the present definition4 ,,end ,,of ,,the ,,first ,,volume4 ,,footnotes3 #a ,the ,grecian literature of this writer will now prove of real utility2 and the graces and the sublimities of ,,plato will soon be familiarised to the ,english reader1 by a hand that ,i am persuaded will not appear inferior to his great original4 ,let me also be permitted to recommend his version of ,,plotinus on ,,the ,,beautiful4 #b i4e4 ,capable of parts4 #c i4e4 ,not capable of parts4 #d ,dr4 ,young1 in his ,night ,thoughts4 #e ,see book the second1 of ,aristotle's ,metaphysics4 #f ,ennead vi4 lib4 vii4 #g ,in his commentary on the #b;d1 #abth1 and #acth books of ,aristotle's ,metaphysics1 page #fj4 ,a ,latin translation only of this invaluable work is extant2 but ,i have fortunately a copy in my possession1 with the version every where corrected by the learned ,thomas ,gale1 and with large extracts from the ,greek4 #h ,see ,proclus on ,plato's ,theology1 p4 #bbf4 #i ,ennead vi4 lib4 #f4 #aj ,in giving monadic number a subsistence in opinion1 ,i have followed the distribution of ,proclus1 in the conclusion of his comment on a point2 and1 ,i think1 not without sufficient reason4 ,for since monadic numbers are more immaterial than geometrical lines and figures1 they must have a more immaterial subsistence4 ,but as they are correspondent to matter1 they cannot reside in the essential reasons of the soul2 nor can they subsist in the phantasy1 because they are superior to geometrical figures4 ,it remains1 therefore1 that we must place them between or cogitation1 and the phantasy2 and this middle situation is that of opinion4 ,for cogitation1 which ,plato defines1 in his ,sophista1 to be an inward discourse1 without voice1 is an energy of the rational soul1 extending itself from propositions to conclusions4 ,and1 according to ,plato1 in the same place1 opinion is the silent affirmation1 or negation of 1 or thought4 ,hence1 says he1 8opinion is the conclusion of cogitation2 but imagination1 the mutual mixture of sense and opinion40 ,so that opinion may1 with great propriety1 be said to contain monadic number1 to which it bears the proportion of matter4 ,and hence the reason is obvious1 why the ,pythagoreans called the duad opinion4 #aa 1 1 4 ,syrian4 in ,meta4 ,aristot4 p4 #aac4 ,gr4 i4e4 7,according to the ,pythagoreans7 8the immortal gods and earth-born men1 call the venerable decad1 immutable and unwearied40 #ab 4 ,vid4 ,syrian4 in ,arist4 ,meta4 p4 #he4 ,gr4 #ac 1 4 ,syrian4 in eodem loco4 #ad 1 4 4 ,jamb4 in ,nicomach4 ,arith4 p4 #aa4 #ae ,in his ,mathematical ,lectures1 page #dh4 #af ,in ,arithmet4 p4 #bc4 #ag ,in ,aristot4 ,meta4 p4 #aac4 ,gr4 vel #ei4 b4 ,lat4 #ah ,for the tetrad contains all numbers within its nature1 in the manner of an exemplar2 and hence it is1 that in monadic numbers1 #a1 #b1 #c1 #d1 are equal to ten4 #ai ,notes to ,letters on ,mind1 page #hc4 #bj ,this bright light is no other than that of ideas themselves2 which1 when it is once enkindled1 or rather re-kindled in the soul1 becomes the general standard1 and criterion of truth4 ,he who possesses this1 is no longer the slave of opinion2 puzzled with doubts1 and lost in the uncertainties of conjecture4 ,here the fountain of evidence is alone to be found4--,this is the true light1 whose splendors can alone dispel the darkness of ignorance1 and procure for the soul undecaying good1 and substantial felicity4 ,of this ,i am certain1 from my own experience2 and happy is he who acquires this invaluable treasure4 ,but let the reader beware of mixing the extravagancies of modern enthusiasm with this exalted illumination4 ,for this light is alone brought into the mind by science1 patient reflection1 and unwearied meditation3 it is not produced by any violent agitation of spirits1 or extasy of imagination2 for it is far superior to the energies of these3 but it is tranquil and steady1 intellectual and divine4 ,avicenna1 the ,arabian1 was well acquainted with this light1 as is evident from the beautiful description he gives of it1 in the elegant introduction of ,ebn ,tophail1 to the ,life of ,hai ,ebn ,yokdhan4 8,when a man's desires 7says he7 are considerably elevated1 and he is competently well exercised in these speculations1 there will appear to him some small glimmerings of the truth1 as it were flashes of lightning1 very delightful1 which just shine upon him1 and then become extinct4 ,then the more he exercises himself1 the oftener will he perceive them1 till at last he will become so well acquainted with them1 that they will occur to him spontaneously1 without any exercise at all2 and then as soon as he perceives any thing1 he applies himself to the divine essence1 so as to retain some impression of it2 then something occurs to him on a sudden1 whereby he begins to discern the truth in every thing2 till through frequent exercise he at last attains to a perfect tranquillity2 and that which used to appear to him only by fits and starts1 becomes habitual1 and that which was only a glimmering before1 a constant light2 and he obtains a constant and steady knowledge40 ,he who desires to know more concerning this1 and a still brighter light1 that arising from an union with the supreme1 must consult the eighth book of ,plotinus' fifth ,ennead1 and the #gth and #ith of the sixth1 and his book on the ,beautiful1 of which ,i have published a translation4 #ba ,lest the superficial reader should think this is nothing more than declamation1 let him attend to the following argument4 ,if the soul possesses another eye different from that of sense 7and that she does so1 the sciences sufficiently evince71 there must be1 in the nature of things1 species accommodated to her perception1 different from feasible forms4 ,for if our intellect speculates things which have no real subsistence1 such as ,mr4 ,locke's ideas1 its condition must be much more unhappy than that of the sensitive eye1 since this is co-ordinated to beings2 but intellect would speculate nothing but illusions4 ,now1 if this be absurd1 and if we possess an intellectual eye1 which is endued with a visive power1 there must be forms correspondent and conjoined with its vision2 forms immoveable1 indeed1 by a corporeal motion1 but moved by an intellectual energy4 #bb ,the present section contains an illustration of almost all the first book of ,aristotle's last ,analytics4 ,i have for the most part followed the accurate and elegant paraphrase of ,themistius1 in the execution of this design1 as the learned reader will perceive3 but ,i have likewise everywhere added elucidations of my own1 and endeavoured to render this valuable work intelligible to the thinking mathematical reader4 #bc ,see the twenty-eighth proposition of the first book of ,euclid's ,elements4 #bd ,we are informed by ,simplicius1 in his ,commentary on ,aristotle's third ,category of ,relation1 8that though the quadrature of the circle seems to have been unknown to ,aristotle1 yet1 according to ,jamblichus1 it was known to the ,pythagoreans1 as appears from the sayings and demonstrations of ,sextus ,pythagoricus1 who received 7says he7 by succession1 the art of demonstration2 and after him ,archimedes succeeded1 who invented the quadrature by a line1 which is called the line of ,nicomedes4 ,likewise1 ,nicomedes attempted to square the circle by a line1 which is properly called 1 or the quadrature4 ,and ,apollonius1 by a certain line1 which he calls the sister of the curve line1 similar to a cockle1 or tortoise1 and which is the same with the quadratix of ,nicomedes4 ,also ,carpus wished to square the circle1 by a certain line1 which he calls simply formed from a twofold motion4 ,and many others1 according to ,jamblichus1 have accomplished this undertaking in various ways40 ,thus far ,simplicius4 ,in like manner1 ,boethius1 in his ,commentary on the same part of ,aristotle's ,categories 7p4 #aff47 observes1 that the quadrature of the circle was not discovered in ,aristotle's time1 but was found out afterwards2 the demonstration of which 7says he7 because it is long1 must be omitted in this place4 ,from hence it seems very probable1 that the ancient mathematicians applied themselves solely to squaring the circle geometrically1 without attempting to accomplish this by an arithmetical calculation4 ,indeed1 nothing can be more ungeometrical than to expect1 that if ever the circle be squared1 the square to which it is equal must be commensurable with other known rectilineal spaces2 for those who are skilled in geometry know that many lines and spaces may be exhibited with the greatest accuracy1 geometrically1 though they are incapable of being expressed arithmetically1 without an infinite series4 ,agreeable to this1 ,tacquet well observes 7in lib4 ii4 ,geom4 ,pract4 p4 #hg47 8,denique admonendi hic sunt1 qui geometri1 non satis periti1 sibi persuadent ad quadraturam necessarium esse1 ut ratio line circularis ad rectam1 aut circuli ad quadratum in numeris exhibeatur4 ,is sane error valde crassus est1 et indignus geometr1 quamvis enim irrationalis esset ea proportio1 modo in rectis lineis exhibeatur1 reperta erat quadratura40 ,and that this quadrature is possible geometrically1 was not only the opinion of the above mentioned learned and acute geometrician1 but likewise of ,wallis and ,barrow2 as may be seen in the ,mechanics of the former1 p4 #eag and in the ,mathematical ,lectures of the latter1 p4 #aid4 ,but the following discovery will1 ,i hope1 convince the liberal geometrical reader1 that the quadrature of the circle may be obtained by means of a circle and right-line only1 which we have no method of accomplishing by any invention of the ancients or moderns4 ,at least this method1 if known to the ancients1 is now lost1 and though it has been attempted by many of the moderns1 it has not been attended with success4 ,in the circle g o e f1 let g o be the quadrantal arch1 and the right-line g x its tangent4 ,then conceive that the central point a flows uniformly along the radius a e1 infinitely produced2 and that it is endued with an uniform impulsive power4 ,let it likewise be supposed1 that during its flux1 radii emanate from it on all sides1 which enlarge themselves in proportion to the distance of the point a from its first situation4 ,this being admitted1 conceive that the point a by its impulsive power1 through the radii a n1 a m1 c4 acting every where equally on the arch g o1 impells it into its equal tangent arch g r4 ,and when1 by its uniform motion along the infinite line a 1 it has at the same time arrived at b1 the centre of the arch g r1 let it impel in a similar manner the arch g r1 into its equal tangent arch g s1 by acting every where equally through radii equal to b r4 ,now1 if this be conceived to take place infinitely 7since a circular line is capable of infinite remission7 the arch g o will at length be unbent into its equal1 the tangent line g x2 and the extreme point o1 will describe by such a motion of unbending a circular line o x4 ,for since the same cause1 acting every where similarly and equally1 produces every where similar and equal effects2 and the arch g o1 is every where equally remitted or unbent1 it will describe a line similar in every part4 ,now1 on account of the simplicity of the impulsive motion1 such a line must either be straight or circular2 for there are only three lines every where similar1 i4 e4 the right and circular line1 and the cylindric helix2 but this last1 as ,proclus well observes in his following ,commentary on the fourth definition1 is not a simple line1 because it is generated by two simple motions1 the rectilineal and circular4 ,but the line which bounds more than two equal tangent arches cannot be a right line1 as is well known to all geometricians2 it is therefore a circular line4 ,it is likewise evident1 that this arch o x is concave towards the point g3 for if not1 it would pass beyond the chord o x1 which is absurd4 ,and again1 no arch greater than the quadrant can be unbent by this motion3 for any one of the radii1 as a p beyond g o1 has a tendency from1 and not to the tangent g x1 which last is necessary to our hypothesis4 ,now if we conceive another quadrantal arch of the circle g o e f1 that is g y1 touching the former in g to be unbent in the same manner1 the arch x y shall be a continuation of the arch x o2 for if x be drawn perpendicular to x g1 as in the figure1 it shall be a tangent in x to the equal arches y x1 x o2 because it cannot fall within either1 without making the sine of some one of the equal arches1 equal to the right-line x g1 which would be absurd4 ,and hence we may easily infer1 that the centre of the arch y x o1 is in the tangent line x g4 ,hence too1 we have an easy method of finding a tangent right-line equal to a quadrantal arch3 for having the points y1 o given1 it is easy to find a third point1 as s2 and then the circle passing through the three points o1 s1 y1 shall cut off the tangent x g1 equal to the quadrantal arch g o4 ,and the point s may be speedily obtained1 by describing the arch g s with a radius1 having to the radius a g the proportion of #f to #d2 for then g s is the sixth part of its whole circle1 and is equal to the arch g o4 ,and thus1 from this hypothesis1 which1 ,i presume1 may be as readily admitted as the increments and decrements of lines in fluxions1 the quadrature of the circle may be geometrically obtained2 for this is easily found1 when a right-line is discovered equal to the periphery of a circle4 ,i am well aware the algebraists will consider it as useless1 because it cannot be accommodated to the farrago of an arithmetical calculation2 but ,i hope the lovers of the ancient geometry will deem it deserving an accurate investigation2 and if they can find no paralogism in the reasoning1 will consider it as a legitimate demonstration4 ,illustration ,illustration #be ,axioms have a subsistence prior to that of magnitudes and mathematical numbers1 but subordinate to that of ideas2 or1 in other words1 they have a middle situation between essential and mathematical magnitude4 ,for of the reasons subsisting in soul1 some are more simple and universal1 and have a greater ambit than others1 and on this account approach nearer to intellect1 and are more manifest and known than such as are more particular4 ,but others are destitute of all these1 and receive their completion from more ancient reasons4 ,hence it is necessary 7since conceptions are then true1 when they are consonant with things themselves7 that there should be some reason1 in which the axiom asserting1 if from equals you take away equals1 c4 is primarily inherent2 and which is neither the reason of magnitude1 nor number1 nor time1 but contains all these1 and every thing in which this axiom is naturally inherent4 ,vide ,syrian4 in ,arith4 ,meta4 p4 #dh4 #bf ,geometry1 indeed1 wishes to speculate the impartible reasons of the soul1 but since she cannot use intellections destitute of imagination1 she extends her discourses to imaginative forms1 and to figures endued with dimension1 and by this means speculates immaterial reasons in these2 and when imagination is not sufficient for this purpose1 she proceeds even to external matter1 in which she describes the fair variety of her propositions4 ,but1 indeed1 even then the principal design of geometry is not to apprehend sensible and external form1 but that interior vital one1 resident in the mirror of imagination1 which the exterior inanimate form imitates1 as far as its imperfect nature will admit4 ,nor yet is it her principal design to be conversant with the imaginative form2 but when1 on account of the imbecility of her intellection1 she cannot receive a form destitute of imagination1 she speculates the immaterial reason in the purer form of the phantasy2 so that her principal employment is about universal and immaterial forms4 ,syrian4 in ,arist4 ,meta4 p4 #di4 #bg ,syrianus1 in his excellent ,commentary on ,aristotle's ,metaphysics1 7which does not so much explain ,aristotle1 as defend the doctrine of ideas1 according to ,plato1 from the apparent if not real opposition of ,aristotle to their existence71 informs us that it is the business of wisdom1 properly so called1 to consider immaterial forms or essences1 and their essential accidents4 ,by the method of resolution receiving the principles of being2 by a divisive and and definitive method1 considering the essences of all things2 but by a demonstrative process1 concluding concerning the essential properties which substances contain4 ,hence 7says he7 because intelligible essences are of the most simple nature1 they are neither capable of definition nor demonstration1 but are perceived by a simple vision and energy of intellect alone4 ,but middle essences1 which are demonstrable1 exist according to their inherent properties3 since1 in the most simple beings1 nothing is inherent besides their being4 ,on which account we cannot say that this is their essence1 and that something else2 and hence they are better than definition and demonstration4 ,but in universal reasons1 considered by themselves1 and adorning a sensible nature1 essential accidents supervene2 and hence demonstration is conversant with these4 ,but in material species1 individuals1 and sensibles1 such things as are properly accidents are perceived by the imagination1 and are present and absent without the corruption of their subjects4 ,and these again being worse than demonstrable accidents1 are apprehended by signs1 not indeed by a wise man1 considered as wise1 but perhaps by physicians1 natural philosophers1 and all of this kind4 #bh ,see ,note to ,chap4 i4 ,book i4 of the ensuing ,commentaries4 #bi ,page #bbg4 #cj ,page #bej4 #ca ,methodus hc cum algebr specios facilitate contendit1 evidenti vero et demonstrationum eleganti eam longe superare videtur3 ut abunde constabit1 si quis conferat hanc ,apollonii doctrinam de ,sectione ,rationis cum ejusdem ,problematis ,analysi ,algebraic1 quam exhibuit clarissimus ,wallisius1 tom4 ii4 ,operum ,math4 cap4 liv4 p4 #bbj4 #cb ,verum perpendendum est1 aliud esse problema aliqualiter resolutum dare1 quod modis variis1 plerumque fieri potest1 aliud methodo elegantissim ipsum efficere2 ,analysi brevissim et simul perspicu1 ,synthesi concinn et minime operos4 #cc ,in his ,mathematical ,lectures1 p4 #dd4 #cd ,lib4 iv4 #ce ,lib4 i4 p4 #cj4 #cf ,in ,theteto4 #cg ,in his most excellent work on ,abstinence1 lib4 i4 p4 #bb1 c4 #ch ,see the ,excerpta of ,ficinus from ,proclus1 on the first ,alcibiades of ,plato2 his ,latin version only of which is extant4 ,ficini ,opera1 tom4 ii4 #ci ,marinus1 the author of the ensuing life1 was the disciple of ,proclus2 and his successor in the ,athenian school4 ,his philosophical writings were not very numerous1 and have not been preserved4 ,a commentary ascribed to him1 on ,euclid's data1 is still extant2 but his most celebrated work1 appears to have been1 the present life of his master4 ,it is indeed in the original elegant and concise2 and may be considered as a very happy specimen of philosophical biography4 ,every liberal mind must be charmed and elevated with the grandeur and sublimity of character1 with which ,proclus is presented to our view4 ,if compared with modern philosophical heroes1 he appears to be a being of a superior order2 and we look back with regret on the glorious period1 so well calculated for the growth of the philosophical genius1 and the encouragement of exalted merit4 ,we find in his life1 no traces of the common frailties of depraved humanity2 no instances of meanness1 or instability of conduct3 but he is uniformly magnificent1 and constantly good4 ,i am well aware that this account of him will be considered by many as highly exaggerated2 as the result of weak enthusiasm1 blind superstition1 or gross deception3 but this will never be the persuasion of those1 who know by experience what elevation of mind and purity of life the ,platonic philosophy is capable of procuring2 and who truly understand the divine truths contained in his works4 ,and the testimony of the multitude1 who measure the merit of other men's characters by the baseness of their own1 is surely not to be regarded4 ,i only add1 that our ,philosopher flourished #dab years after ,christ1 according to the accurate chronology of ,fabricius2 and ,i would recommend those who desire a variety of critical information concerning ,proclus1 to the ,prolegomena prefixed by that most learned man to his excellent ,greek and ,latin edition of this work1 printed at ,london in #agjc4 #dj ,plato in ,phdro4 ,meminit et ,plutarch4 ,,viii4 ,sympos4 ,suidas in 4 ,fabricius4 #da ,for a full account of the distribution of the virtues according to the ,platonists1 consult the sentences of ,porphyry1 and the ,prolegomena of ,fabricius to this work4 #db ,see the sixth book of his ,republic1 and the ,epinomis4 #dc ,we are informed by ,fabricius1 that the ,platonic ,olympiodorus in his ,,ms4 ,commentary on the ,alcibiades of ,plato1 divides the orders of the ,gods1 into 1 or super-mundane1 which are separate from all connection with body2 and into 1 or mundane4 ,and that of these1 some are 1 or celestial1 others 1 or1 or etherial1 or 1 fiery1 others 1 or aerial1 others 1 or watry1 others 1 or earthly2 and others 1 or subterranean4 ,but among the terrestrial1 some are 1 or governors of climates1 others 1 or rulers over cities1 and others lastly 1 or governors of houses4 #dd ,this epithet is likewise ascribed by ,onomacritus to the ,moon1 as may be seen in his hymn to that deity2 and the reason of which we have given in our notes to that hymn4 #de ,divine visions1 and extraordinary circumstances1 may be fairly allowed to happen to such exalted geniuses as ,proclus2 but deserve ridicule when ascribed to the vulgar4 #df ,what glorious times6 when it was considered as an extraordinary circumstance for a teacher of rhetoric to treat a noble and wealthy pupil as his domestic4 ,when we compare them with the present1 we can only exclaim1 ,o tempora6 ,o mores6 ,philosophy sunk in the ruins of ancient ,greece and ,rome4 #dg ,fabricius rightly observes1 that this ,olympiodorus is not the same with the ,philosopher of that name1 whose learned commentaries1 on certain books of ,plato1 are extant in manuscript1 in various libraries4 ,as in these1 not only ,proclus himself1 but ,damascius1 who flourished long after ,proclus1 is celebrated4 #dh ,concerning the various mathematicians of this name1 see ,fabricius in ,bibliotheca ,grca4 #di ,the word in the original is 1 which ,fabricius rightly conjectures has in this place a more extensive signification than either ,logic1 or ,rhetoric3 but ,i must beg leave to differ from that great critic1 in not translating it simply philosophical1 as ,i should rather imagine1 ,marinus intended to confine it to that part of ,aristotle's works1 which comprehends only logic1 rhetoric1 and poetry4 ,for the verb 1 or to learn1 which ,marinus uses on this occasion1 cannot with propriety be applied to the more abstruse writings of ,aristotle4 #ej ,hence ,proclus was called1 by way of eminence1 1 or the ,platonic ,successor4 #ea ,concerning ,polletes1 see ,suidas2 and for ,melampodes1 consult ,fabricius in ,bibliotheca ,grca4 #eb ,this ,syrianus was indeed a most excellent philosopher1 as we may be convinced from his commentary on the metaphysics of ,aristotle1 a ,latin translation only of which1 by one ,hieronimus ,bagolinus1 was published at ,venice in #aeeh4 ,the ,greek is extant1 according to ,fabricius1 in many of the ,italian libraries1 and in the ,johannean library at ,hamburg4 ,according to ,suidas1 he writ a commentary on the whole of ,homer in six books2 on ,plato's politics1 in four books2 and on the consent of ,orpheus1 ,pythagoras1 and ,plato1 with the ,chaldean ,oracles1 in ten books4 ,all these are unfortunately lost2 and the liberal few1 are by this means deprived of treasures of wisdom1 which another philosophical age1 in some distant revolution1 is alone likely to produce4 #ec ,socrates1 in the #fth book of ,plato's ,republic1 says1 that from great geniuses nothing of a middle kind must be expected2 but either great good1 or great evil4 #ed ,the reader will please to take notice1 that this great man is not the same with ,plutarch the biographer1 whose works are so well known2 but an ,athenian philosopher of a much later period4 #ee ,aristotle's philosophy1 when compared with the discipline of ,plato is1 ,i think1 deservedly considered in this place as bearing the relation of the proteleia to the epopteia in sacred mysteries4 ,now the proteleia1 or things previous to perfection1 belong to the initiated1 and the mystics2 the former of whom were introduced into some lighter ceremonies only3 but the mystics1 were permitted to be present with certain preliminary and lesser sacred concerns4 ,on the other hand1 the epopt were admitted into the sanctuary of the greater sacred rites2 and became spectators of the symbols1 and more interior ceremonies4 ,aristotle indeed appears to be every where an enemy to the doctrine of ideas1 as understood by ,plato2 though they are doubtless the leading stars of all true philosophy4 ,however1 the great excellence of his works1 considered as an introduction to the divine theology of ,plato1 deserves the most unbounded commendation4 ,agreeable to this1 ,damascius informs us that ,isidorus the philosopher1 8when he applied himself to the more holy philosophy of ,aristotle1 and saw that he trusted more to necessary reasons than to his own proper sense1 yet did not entirely employ a divine intellection1 was but little solicitous about his doctrine3 but that when he had tasted of ,plato's conceptions1 he no longer deigned to behold him in the language of ,pindar4 ,but hoping he should obtain his desired end1 if he could penetrate into the sanctuary of ,plato's mind1 he directed to this purpose the whole course of his application40 ,photii ,bibliotheca4 p4 #ajcd4 #ef according to the oracle4 #eg ,nothing is more celebrated by the ancients than that strict friendship which subsisted among the ,pythagoreans2 to the exercise of which they were accustomed to admonish each other1 not to divide the god which they contained1 as ,jamblichus relates1 lib4 i4 c4 #cc4 ,de ,vita ,pythagor4 ,indeed1 true friendship can alone subsist in souls1 properly enlightened with genuine wisdom and virtue2 for it then becomes an union of intellects1 and must consequently be immortal and divine4 #eh ,pythagoras1 according to ,damascius1 said1 that friendship was the mother of all the political virtues4 #ei ,a genuine modern will doubtless consider the whole of ,proclus' religious conduct as ridiculously superstitious4 ,and so1 indeed1 at first sight1 it appears2 but he who has penetrated the depths of ancient wisdom1 will find in it more than meets the vulgar ear4 ,the religion of the ,heathens1 has indeed1 for many centuries1 been the object of ridicule and contempt3 yet the author of the present work is not ashamed to own1 that he is a perfect convert to it in every particular1 so far as it was understood and illustrated by the ,pythagoric and ,platonic philosophers4 ,indeed the theology of the ancient1 as well as of the modern vulgar1 was no doubt full of absurdity2 but that of the ancient philosophers1 appears to be worthy of the highest commendations1 and the most assiduous cultivation4 ,however1 the present prevailing opinions1 forbid the defence of such a system2 for this must be the business of a more enlightened and philosophic age4 ,besides1 the author is not forgetful of ,porphyry's destiny1 whose polemical writings were suppressed by the decrees of emperors2 and whose arguments in defence of his religion were so very futile and easy of solution1 that1 as ,st4 ,hierom informs us1 in his preface on ,daniel1 ,eusebius answered him in twenty-five1 and ,apollinaris in thirty volumes6 #fj ,see ,proclus on ,plato's ,politics1 p4 #cii4 ,instit4 ,theolog4 num4 #aif2 and the extracts of ,ficinus from ,proclus's commentary on the first ,alcibiades1 p4 #bdf4 c4 #fa ,alluding to the beautiful description given of ,ulysses1 in the #c;d book of the ,iliad1 v4 #bbb4 4 ,which is thus elegantly paraphrased by ,mr4 ,pope4 ,but when he speaks1 what elocution flows6 ,soft as the fleeces of descending snows ,the copious accents fall1 with easy art2 ,melting they fall1 and sink into the heart6 c4 #fb ,concerning ,domninus1 see ,photius and ,suidas from ,damascius in his ,life of ,isidorus4 #fc ,nicephorus1 in his commentary on ,synesius de ,insomniis1 p4 #efb4 informs us1 that the hecatic orb1 is a golden sphere1 which has a sapphire stone included in its middle part1 and through its whole extremity1 characters and various figures4 ,he adds1 that turning this sphere round1 they perform invocations1 which they call ,jyng4 ,thus too1 according to ,suidas1 the magician ,julian of ,chaldea1 and ,arnuphis the ,egyptian1 brought down showers of rain1 by a magical power4 ,and by an artifice of this kind1 ,empedocles was accustomed to restrain the fury of the winds2 on which account he was called 1 or a chaser of winds4 #fd ,no opinion is more celebrated1 than that of the metempsychosis of ,pythagoras3 but perhaps1 no doctrine is more generally mistaken4 ,by most of the present day it is exploded as ridiculous2 and the few who retain some veneration for its founder1 endeavour to destroy the literal1 and to confine it to an allegorical meaning4 ,by some of the ancients this mutation was limited to similar bodies3 so that they conceived the human soul might transmigrate into various human bodies1 but not into those of brutes2 and this was the opinion of ,hierocles1 as may be seen in his comment on the ,golden ,verses4 ,but why may not the human soul become connected with subordinate as well as with superior lives1 by a tendency of inclination8 ,do not similars love to be united2 and is there not in all kinds of life1 something similar and common8 ,hence1 when the affections of the soul verge to a baser nature1 while connected with a human body1 these affections1 on the dissolution of such a body1 become enveloped as it were1 in a brutal nature1 and the rational eye1 in this case1 clouded with perturbations1 is oppressed by the irrational energies of the brute1 and surveys nothing but the dark phantasms of a degraded imagination4 ,but this doctrine is vindicated by ,proclus with his usual subtilty1 in his admirable commentary on the ,timus1 lib4 v4 p4 #cbi1 as follows1 8,it is usual1 says he1 to enquire how souls can descend into brute animals4 ,and some1 indeed1 think that there are certain similitudes of men to brutes1 which they call savage lives3 for they by no means think it possible that the rational essence can become the soul of a savage animal4 ,on the contrary1 others allow it may be sent into brutes1 because all souls are of one and the same kind2 so that they may become wolves and panthers1 and ichneumons4 ,but true reason1 indeed1 asserts that the human soul way be lodged in brutes1 yet in such a manner1 as that it may obtain its own proper life1 and that the degraded soul may1 as it were1 be carried above it1 and be bound to the baser nature1 by a propensity and similitude of affection4 ,and that this is the only mode of insinuation1 we have proved by a multitude of reasons1 in our commentaries on the ,phdrus4 ,but if it is requisite to take notice1 that this is the opinion of ,plato1 we add1 that in his politics1 he says1 that the soul of ,thersites assumed an ape1 but not the body of an ape3 and in the ,phdrus1 that the soul descends into a savage life1 but not into a savage body2 for life is conjoined with its proper soul4 ,and in this place he says it is changed into a brutal nature3 for a brutal nature is not a brutal body1 but a brutal life40 #fe ,pericles ,lydus1 a ,stoic philosopher4 #ff ,vide ,pausan4 lib4 i4 ,atticorum1 cap4 #ba4 et #bj4 #fg ,he means the ,christians4 #fh ,proclus was born in the year of ,christ #dab1 on the #fth of the ,ides of ,february4 ,but1 for the sake of the astrologers1 ,i have subjoined the following figure from the ,prolegomena of ,fabricius to this life3 and though ,i am not skilled in the art myself1 ,i am persuaded1 from the arguments of ,plotinus1 that it contains many general truths2 but when made subservient to particulars1 is liable to great inaccuracy and error4 ,in short1 its evidence is wholly of a physiognomic nature2 for such is the admirable order and connection of things1 that throughout the universe1 one thing is signified by another1 and wholes are after a manner contained in their parts4 ,so that the language of the obscure and profound ,heraclitus is perfectly just1 when he says1 8,you must connect the perfect and the imperfect1 the agreeing and the disagreeing1 the consonant and the dissonant1 and out of one all things1 and out of all things one40 ,illustration3 ,a ,scheme of the situation of the ,stars1 such as it was at ,byzantium1 when the philosopher ,proclus was born4 #fi ,it was formerly the custom of almost all nations1 to have their burial places in the suburbs1 and not in the city itself4 #gj ,this eclipse happened1 according to ,fabricius1 in ,a4 ,c4 #dhd4 #ai ,cal4 ,feb4 at sun-rise4 #ga ,all the ancient theologists1 among whom ,plato holds a distinguished rank1 affirmed that the soul was of a certain middle nature and condition between intelligibles and sensibles3 agreeable to which doctrine1 ,plotinus divinely asserts that she is placed in the horizon1 or in the boundary and isthmus1 as it were1 of eternal and mortal natures2 and hence1 according to the ,magi1 she is similar to the moon1 one of whose parts is lucid1 but the other dark4 ,now the soul1 in consequence of this middle condition1 must necessarily be the receptacle of all middle energies1 both vital and gnostic2 so that her knowledge is inferior to the indivisible simplicity of intellectual comprehension1 but superior to the impulsive perceptions of sense4 ,hence the mathematical genera and species reside in her essence1 as in their proper and natural region2 for they are entirely of a middle nature1 as ,proclus proves in this and the sixth following chapter4 ,but this doctrine of ,plato's1 originally derived from ,brontinus and ,archytas1 is thus elegantly explained by that philosopher1 in the concluding part of the sixth book of his ,republic4 8,socrates1 know then1 they are1 as we say1 two 7the ,good itself1 and the ,sun17 and that the one reigns over the intelligible world1 but the other over the visible1 not to say the heavens1 lest ,i should deceive you by the name4 ,you comprehend then1 these two orders of things1 ,i mean the visible and the intelligible8--,glauco4 ,i do4--,socrates4 ,continue this division then1 as if it were a line divided into two unequal segments2 and each part again1 i4 e4 the sensible and intelligible1 divided after a similar manner1 and you will have evidence and obscurity placed by each other4 ,in the visible segment1 indeed1 one part will contain images4 ,but ,i call images1 in the first place1 shadows2 afterwards1 the resemblances of things appearing in water1 and in dense1 smooth1 and lucid bodies1 and every thing of this kind1 if you apprehend me8--,glauco4 ,i apprehend you4--,socrates4 ,now conceive that the other section comprehends the things1 of which these images are nothing more than similitudes1 such as the animals around us1 together with plants1 and whatever is the work of nature and art4--,glauco4 ,i conceive it4--,socrates4 ,do you consider this section then1 as divided into true and false8 ,and that the hypothesis of opinion is to the knowledge of science1 as a resemblance to its original8--,glauco4 ,i do1 very readily4--,socrates4 ,now then1 consider how the section of the intelligible is to be divided4--,glauco4 ,how8--,socrates4 ,thus3 one segment is that which the soul enquires after1 using the former divisions as images1 and compelled to proceed from hypotheses1 not to the principle1 but to the conclusion4 ,the other is that which employs the cogitative power of the soul1 as she proceeds from an hypothesis to a principle no longer supposed1 and1 neglecting images1 advances through their obscurity into the light of ideas themselves4--,glauco4 ,i do not1 in this1 sufficiently understand you4--,socrates4 ,but again1 for you will more easily understand me from what has been already premised4 ,i think you are not ignorant1 that those who are conversant in geometry1 arithmetic1 and the like1 suppose even and odd1 together with various figures1 and the three species of angles1 and other things similar to these1 according to each method of proceeding4 ,now1 having established these1 as hypotheses sufficiently known1 they conceive that no reason is to be required for their position3 but beginning from these1 they descend through the rest1 and arrive at last1 at the object of their investigation4--,glauco4 ,this ,i know perfectly well4--,socrates4 ,this also you know1 that they use visible forms1 and make them the subject of their discourse1 at the same time not directing their intellect to the perception of these1 but to the originals they resemble2 ,i mean the square itself1 and the diameter itself2 and not to the figures they delineate4 ,and thus1 other forms1 which are represented by shadows and images in water1 are employed by them1 merely as resemblances1 while they strive to behold that which can be seen by cogitation alone4--,glauco4 ,you speak the truth4--,socrates4 ,this is what ,i called above a species of the intelligible1 in the investigation of which1 the soul was compelled to use hypotheses2 not ascending to the principle1 as incapable of rising above hypotheses1 but using the images formed from inferior objects1 to a similitude of such as are superior1 and which are so conceived and distinguished by opinion1 as if they perspicuously contributed to the knowledge of things themselves4--,glauco4 ,i understand indeed1 that you are speaking of the circumstances which take place in geometry1 and her kindred arts4--,socrates4 ,understand now1 that by the other section of the intelligible1 ,i mean that which reason herself reaches1 by her power of demonstrating1 when no longer esteeming hypotheses for principles1 but receiving them in reality for hypotheses1 she uses them as so many steps and handles in her ascent1 until she arrives at that which is no longer hypothetical1 the principle of the universe2 and afterwards descending1 holding by ideas which adhere to the principle1 she arrives at the conclusion1 employing nothing sensible in her progress1 but proceeding through ideas1 and in these at last terminating her descent4--,glauco4 ,i understand you1 but not so well as ,i desire3 for you seem to me to propose a great undertaking4 ,you endeavour1 indeed1 to determine that the portion of true being and intelligible1 which we speculate by the science of demonstration1 is more evident than the discoveries made by the sciences called arts2 because in the first hypotheses are principles1 and their masters are compelled to employ the eye of cogitation1 and not the perceptions of the senses4 ,yet1 because they do not ascend to the principle1 but investigate from hypotheses1 they seem to you not to have intelligence concerning these1 though they are intelligible1 through the light of the principle4 ,but you seem to me to call the habit of reasoning on geometrical and the like concerns1 cogitation1 rather than intelligence1 as if cogitation held the middle situation between opinion and intellect4--,socrates4 ,you understand me sufficiently well4 ,and again3 with these four proportions take these four corresponding affections of the soul3 with the highest intelligence2 with the second cogitation2 against the third set opinion2 and against the fourth assimilation1 or imagination4 ,besides this1 establish them in the order of alternate proportion1 so that they may partake of evidence1 in the same manner as their corresponding objects participate of reality40 ,i have taken the liberty of translating this fine passage differently from both ,petvin and ,spens2 because they have neglected to give the proper meaning of the word 1 or cogitation1 the former translating it mind1 and the eye of the mind1 and by this means confounding it with intellect2 and the latter calling it understanding4 ,but it is certain that ,plato1 in this place1 ranks intellect as the first1 on account of the superior evidence of its perceptions2 in the next place1 cogitation2 in the third1 opinion2 and in the fourth1 imagination4 ,however1 the reader will please to remember1 that by 1 or cogitation1 in the present work1 is understood that power of the soul which reasons from premises to conclusions1 and whose syllogistic energy1 on active subjects1 is called prudence2 and on such as are speculative1 science4 ,but for farther information concerning its nature1 see the dissertation prefixed to this work1 and the following fifth chapter4 #gb ,these two principles1 bound and infinite1 will doubtless be considered by the unthinking part of mankind1 as nothing more than general terms1 and not as the most real of beings4 ,however1 an accurate contemplation of the universe1 will convince every truly philosophic mind of their reality4 ,for the heavens themselves1 by the coherence and order of their parts1 evince their participation of bound4 ,but by their prolific powers1 and the unceasing revolutions of the orbs they contain1 they demonstrate their participation of infinity4 ,and the finite and perpetually abiding forms with which the world is replete1 bear a similitude to bound3 while1 on the contrary1 the variety of particulars1 their never-ceasing mutation1 and the connection of more and less in the communion of forms1 represents an image of infinity4 ,add too1 that every natural species1 by its form is similar to bound2 but by its matter1 to infinity4 ,for these two1 form and matter1 depend on bound and infinity1 and are their ultimate progressions4 ,and each of these1 indeed1 participates of unity2 but form is the measure and bound of matter1 and is more one4 ,but matter is in capacity all things1 because it subsists by an emanation from the first capacity1 or the infinite itself4 #gc ,of human disciplines1 those alone deserve to be called sciences which use no hypotheses1 which resolve things into their principles1 which are conversant with true being1 and elevate us to ideas themselves4 ,dialectic is wholly of this kind 7,i mean the dialectic of ,plato72 for this alone uses no suppositions1 but1 neglecting shadows and images1 raises us1 by a sublime investigation1 to the principle of the universe2 and on this account1 deserves to be called the very apex of disciplines4 ,but we must not imagine1 that by the word dialectic here1 is meant logic1 or any part of logic1 or that method of disputation1 by which we fabricate probable reasons2 but we must conceive it as signifying a discipline1 endued with the greatest acuteness2 neglecting all hypotheses1 truly soaring to primary causes1 and ultimately reposing in their contemplation4 ,plotinus has given us most happy specimens of this method1 in his books on the genera of being4 #gd ,see note to the first chapter4 #ge ,i would particularly recommend this chapter to modern mathematicians1 most of whom1 ,i am afraid1 have never considered whether or not the subjects of their speculation have any real subsistence3 though it is surely an enquiry worthy the earnest attention of every liberal mind4 ,for if the objects of mathematical investigation are merely imaginary1 ,i mean the point without parts1 the line without breadth1 c4 the science1 founded on these false principles1 must of course be entirely delusive4 ,indeed1 an absolutely true conclusion1 can never flow from an erroneous principle1 as from its cause3 as the stream must always participate of its source4 ,i mean such a conclusion as is demonstrated by the proper cause1 1 ' 1 says ,aristotle1 in his first ,analytics2 that is1 a syllogism from false principles will not prove the why1 but only simply that it is3 indeed it can only simply prove that it is1 to him who admits the false propositions2 because he who allows the premises1 cannot deny the conclusion1 when the syllogism is properly constructed4 ,thus we way syllogize in the first figure1 ,every thing white1 is an animal3 ,every bird is white3 ,therefore1 ,every bird is an animal4 ,and the conclusion will be true1 though the major and minor terms are false2 but then these terms are not the causes of the conclusion1 and we have an inference without a proof4 ,in like manner1 if mathematical species are delusive and fictitious1 the conclusions deduced from them as principles1 are merely hypothetical1 and not demonstrative4 #gf ,aristotle1 in his last ,analytics4 ,the reader will please to observe1 that the whole force of this nervous1 accurate1 and elegant reasoning1 is directed against ,aristotle2 who seems unfortunately to have considered1 with the moderns1 that mathematical species subsist in the soul1 by an abstraction from sensibles4 ,see the preceding ,dissertation4 #gg ,viz4 #a1 #b1 #d1 #h1 #c1 #i1 #bg4 ,concerning which1 see lib4 iii4 of ,proclus's excellent ,commentary on the ,timus4 #gh ,plato frequently1 both in the ,meno and elsewhere1 shews that science is ,reminiscence2 and ,i think not without the strongest reason4 ,for since the soul is immaterial1 as we have demonstrated in the dissertation to this work1 she must be truly immortal1 i4 e4 both a parte ante1 a parte post4 ,that she must be eternal1 indeed1 with respect to futurity1 if immaterial1 is admitted by all2 and we may prove1 with ,aristotle1 in his first book de ,clo1 that she is immortal1 likewise a parte ante1 as follows4 ,every thing without generation1 is incorruptible1 and every thing incorruptible1 is without generation3 for that which is without generation1 has a necessity of existing infinitely a parte ante 7from the hypothesis72 and therefore1 if it possesses a capacity of being destroyed1 since there is no greater reason why it should be corrupted now1 rather than in some former period1 it is endued with a capacity of being destroyed and ceasing to be1 in every instant of infinite time1 in which it necessarily is4 ,in like manner1 that which is incorruptible1 has a necessity of existing infinitely a parte post2 therefore1 if it possesses a capacity of being generated1 since there is no greater reason why it should be generated now rather than afterwards1 it possesses a capacity of being generated1 in every instant of time1 in which it necessarily is4 ,if then the soul is essentially immortal1 with respect to the past and future circulations of time2 and if she is replete with forms or ideas of every kind1 as we have proved in the dissertation1 she must1 from her circulating nature1 have been for ever conversant in alternately possessing and losing the knowledge of these4 ,now1 the recovery of this knowledge by science1 is called by ,plato1 reminiscence2 and is nothing more than a renewed contemplation of those divine forms1 so familiar to the soul1 before she became involved in the dark vestment of an earthly body4 ,so that we may say1 with the elegant ,maximus ,tyrus1 7,disser4 #bh47 8,reminiscence is similar to that which happens to the corporeal eye1 which1 though always endued with a power of vision1 yet darkness sometimes obstructs its passage1 and averts it from the perception of things4 ,art therefore1 approaches1 which though it does not give to the eye the power of vision1 yet removes its impediments1 and affords a free egress to its rays4 ,conceive now1 that our rational soul is such a power of perceiving1 which sees and knows the nature of beings4 ,to this the common calamity of bodies happens1 that darkness spreading round it1 hurries away its aspect1 blunts its sharpness1 and extinguishes its proper light4 ,afterwards1 the art of reason approaches1 which1 like a physician1 does not bring or afford it a new science1 but rouses that which it possesses1 though very slender1 confused1 and unsteady40 ,hence1 since the soul1 by her immersion in body1 is in a dormant state1 until she is roused by science to an exertion of her latent energies2 and yet even previous to this awakening1 since she contains the vivid sparks1 as it were1 of all knowledge1 which only require to be ventilated by the wings of learning1 in order to rekindle the light of ideas1 she may be said in this case to know all things as in a dream1 and to be ignorant of them with respect to vigilant perceptions4 ,hence too1 we may infer that time does not antecede our essential knowledge of forms1 because we possess it from eternity3 but it precedes our knowledge with respect to a production of these reasons into perfect energy4 ,i only add1 that ,i would recommend the liberal ,english reader1 to ,mr4 ,sydenham's excellent translation of ,plato's ,meno1 where he will find a familiar and elegant demonstration of the doctrine of ,reminiscence4 #gi ,concerning this valuable work1 entitled #8 '1 see the ,bibliotheca ,grca of ,fabricius1 vol4 i4 p4 #aah and #dfb1 and in the commentary of ,syrianus on ,aristotle's metaphysics1 p4 #g1 #ga1 #hc1 and #ajh1 the reader will find some curious extracts from this celebrated discourse2 particularly in p4 #hc4 ,syrianus informs us1 8that he who consults this work will find all the orders both of ,monads and ,numbers1 without neglecting one1 fully celebrated 7470 ,there is no doubt1 but that ,pythagoras and his disciples concealed the sublimest truths1 under the symbols of numbers2 of which he who reads and understands the writings of the ,platonists will be fully convinced4 ,hence ,proclus1 in the third book of his excellent commentary on the ,timus1 observes1 8that ,plato employed mathematical terms for the sake of mystery and concealment1 as certain veils1 by which the penetralia of truth might be secluded from vulgar inspection1 just as the theologists made fables1 but the ,pythagoreans symbols1 subservient to the same purpose3 for in images we may speculate their exemplars1 and the former afford us the means of access to the latter40 #hj ,concerning this ,geometric ,number1 in the #hth book of ,plato's ,republic1 than which ,cicero affirms there is nothing more obscure1 see the notes of ,bullialdus to ,theo4 p4 #bib4 #ha ,i am sorry to say1 that this part of the enemies to pure geometry and arithmetic1 are at the present time very numerous2 conceptions of utility in these sciences1 extending no farther than the sordid purposes of a mere animal life4 ,but surely1 if intellect is a part of our composition1 and the noblest part too1 there must be an object of its contemplation2 and this1 which is no other than truth in the most exalted sense1 must be the most noble and useful subject of speculation to every rational being4 #hb ,in the #acth book of his ,metaphysics1 cap4 iii4 #hc ,in4 ,i4 ,de ,partib4 ,animalium1 et in primo ,ethic4 cap4 iii4 #hd ,see more concerning this in the ,dissertation4 #he ,since number is prior to magnitude1 the demonstrations of arithmetic must be more intellectual1 but those of geometry more accommodated to the rational power4 ,and when either arithmetic or geometry is applied to sensible concerns1 the demonstrations1 from the nature of the subjects1 must participate of the obscurity of opinion4 ,if this is the case1 a true mathematician will value those parts of his science most1 which participate most of evidence2 and will consider them as degraded1 when applied to the common purposes of life4 #hf ,this division of the mathematical science1 according to the ,pythagoreans1 which is nearly coincident with that of ,plato1 is blamed by ,dr4 ,barrow in his ,mathematical ,lectures1 p4 #ae4 as being confined within too narrow limits3 and the reason he assigns for so partial a division1 is1 8because1 in ,plato's time1 others were either not yet invented1 or not sufficiently cultivated1 or at least were not yet received into the number of the mathematical sciences40 ,but ,i must beg leave to differ from this most illustrious mathematician in this affair2 and to assert that the reason of so confined a distribution 7as it is conceived by the moderns7 arose from the exalted conceptions these wise men entertained of the mathematical sciences1 which they considered as so many preludes to the knowledge of divinity1 when properly pursued2 but they reckoned them degraded and perverted1 when they became mixed with sensible objects1 and were applied to the common purposes of life4 #hg ,that is1 a right and circular line4 #hh ,i am afraid there are few in the present day1 who do not consider tactics as one of the most principal parts of mathematics2 and who would not fail to cite1 in defence of their opinions1 that great reformer of philosophy1 as he is called1 ,lord ,bacon1 commending pursuits which come home to men's businesses and bosoms4 ,indeed1 if what is lowest in the true order of things1 and best administers to the vilest part of human nature1 is to have the preference1 their opinion is right1 and ,lord ,bacon is a philosopher6 #hi ,by this is to be understood the art new called ,perspective3 from whence it is evident that this art was not unknown to the ancients1 though it is questioned by the moderns4 #ij ,from hence it appears1 that it is doubtful whether ,plato is the author of the dialogue called ,epinomis2 and ,i think it may with great propriety be questioned4 ,for though it bears evident marks of high antiquity1 and is replete with genuine wisdom1 it does not seem to be perfectly after ,plato's manner2 nor to contain that great depth of thought with which the writings of this philosopher abound4 ,fabricius 7in his ,bibliotheca ,grca1 lib4 i4 p4 #bg47 wonders that ,suidas should ascribe this work to a philosopher who distributed ,plato's laws into twelve books1 because it was an usual opinion2 from whence it seems1 that accurate critic had not attended to the present passage4 #ia ,this proximate conjunction of the mathematical sciences1 which ,proclus considers as subordinate to dialectic1 seems to differ from that vertex of science in this1 that the former merely embraces the principles of all science1 but the latter comprehends the universal genera of being1 and speculates the principle of all4 #ib ,in the ,meno4 #ic ,this is certainly the true or philosophical employment of the mathematical science2 for by this means we shall be enabled to ascend from sense to intellect1 and rekindle in the soul that divine light of truth1 which1 previous to such an energy1 was buried in the obscurity of a corporeal nature4 ,but by a contrary process1 ,i mean1 by applying mathematical speculations1 to experimental purposes1 we shall blind the liberal eye of the soul1 and leave nothing in its stead but the darkness of corporeal vision1 and the phantoms of a degraded imagination4 #id ,the design of the present chapter is to prove that the figures which are the subjects of geometric speculation1 do not subsist in external and sensible matter1 but in the receptacle of imagination1 or the matter of the phantasy4 ,and this our philosopher proves with his usual elegance1 subtilty1 and depth4 ,indeed1 it must be evident to every attentive observer1 that sensible figures fall far short of that accuracy and perfection which are required in geometrical definitions3 for there is no sensible circle perfectly round1 since the point from which it is described is not without parts2 and1 as ,vossius well observes1 7de ,mathem4 p4 #d47 there is not any sphere in the nature of things1 that only touches in a point1 for with some part of its superficies it always touches the subjected plane in a line1 as ,aristotle shews ,protagoras to have objected against the geometricians4 ,nor must we say1 with that great mathematician ,dr4 ,barrow1 in his ,mathematical ,lectures1 page #gf1 8that all imaginable geometrical figures1 are really inherent in every particle of matter1 in the utmost perfection1 though not apparent to sense2 just as the effigies of ,csar lies hid in the unhewn marble1 and is no new thing made by the statuary1 but only is discovered and brought to sight by his workmanship1 i4 e4 by removing the parts of matter by which it is overshadowed and involved4 ,which made ,michael ,angelus1 the most famous carver1 say1 that sculpture was nothing but a purgation from things superfluous4 ,for take all that is superfluous1 7says he7 from the wood or stone1 and the rest will be the figure you intend4 ,so1 if the hand of an angel 7at least the power of ,god7 should think fit to polish any particle of matter1 without vacuity1 a spherical superficies would appear to the eyes1 of a figure exactly round2 not as created anew1 but as unveiled and laid open from the disguises and covers of its circumjacent matter40 ,for this would be giving a perfection to sensible matter1 which it is naturally incapable of receiving3 since external body is essentially full of pores and irregularities1 which must eternally prevent its receiving the accuracy of geometrical body1 though polished by the hand of an angel4 ,besides1 what polishing would ever produce a point without parts1 and a line without breadth8 ,for though body may be reduced to the greatest exility1 it will not by this means ever pass into an incorporeal nature1 and desert its triple dimension4 ,since external matter1 therefore1 is by no means the receptacle of geometrical figures1 they must necessarily reside in the catoptric matter of the phantasy1 where they subsist with an accuracy sufficient for the energies of this science4 ,it is true1 indeed1 that even in the purer matter of imagination1 the point does not appear perfectly impartible1 nor the line without latitude3 but then the magnitude of the point1 and the breadth of the line is indefinite1 and they are1 at the same time1 unattended with the qualities of body1 and exhibit to the eye of thought1 magnitude alone4 ,hence1 the figures in the phantasy1 are the proper recipients of that universal1 which is the object of geometrical speculation1 and represent1 as in a mirror1 the participated subsistence of those vital and immaterial forms which essentially reside in the soul4 #ie ,this division is elegantly explained by ,ammonius1 7in ,porphyr4 p4 #ab47 as follows1 8,conceive a seal-ring1 which has the image of some particular person1 for instance1 of ,achilles1 engraved in its seal1 and let there be many portions of wax1 which are impressed by the ring4 ,afterwards conceive that some one approaches1 and perceives all the portions of wax1 stamped with the impression of this one ring1 and keeps the impression of the ring in his mind3 the seal engraved in the ring1 represents the universal1 prior to the many3 the impression in the portions of wax1 the universal in the many3 but that which remains in the intelligence of the beholder1 may be called the universal1 after and posterior to the many4 ,the same must we conceive in genera and species4 ,for that best and most excellent artificer of the world1 possesses within himself the forms and exemplars of all things3 so that in the fabrication of man1 he looks back upon the form of man resident in his essence1 and fashions all the rest according to its exemplar4 ,but if any one should oppose this doctrine1 and assert that the forms of things do not reside with their artificer1 let him attend to the following arguments4 ,the artificer either knows1 or is ignorant of that which he produces3 but he who is ignorant will never produce any thing4 ,for who will attempt to do that1 which he is ignorant how to perform8 since he cannot act from an irrational power like nature1 whose operations are not attended with animadversion4 ,but if he produces any thing by a certain reason1 he must possess a knowledge of every thing which he produces4 ,if1 therefore1 it is not impious to assert1 that the operations of the ,deity1 like those of men1 are attended with knowledge1 it is evident that the forms of things must reside in his essence3 but forms are in the demiurgus1 like the seal in the ring2 and these forms are said to be prior to the many1 and separated from matter4 ,but the species man1 is contained in each particular man1 like the impression of the seal in the wax1 and is said to subsist in the many1 without a separation from matter4 ,and when we behold particular men1 and perceive the same form and effigy in each1 that form seared in our soul1 is said to be after the many1 and to have a posterior generation3 just as we observed in him1 who beheld many seals impressed in the wax from one and the same ring4 ,and this one1 posterior to the many1 may be separated from body1 when it is conceived as not inherent in body1 but in the soul3 but is incapable of a real separation from its subject40 ,we must here1 however1 observe1 that when ,ammonius speaks of the knowledge of the ,deity1 it must be conceived as far superior to ours4 ,for he possesses a nature more true than all essence1 and a perception clearer than all knowledge4 ,and as he produced all things by his unity1 so by an ineffable unity of apprehension1 he knows the universality of things4 #if ,in lib4 vii4 ,metaphys4 #ce #ci4 #ig ,in lib4 iii4 de ,anima1 tex4 #bj4 #ih ,that is1 geometry first speculates the circle delineated on paper1 or in the dust3 but by the medium of the circular figure in the phantasy1 contemplates the circle resident in cogitation2 and by that universal1 or circular reason1 participated in the circle of the phantasy1 frames its demonstrations4 #ii ,in his first ,analytics1 t4 #db4 ,see the ,dissertation to this work4 #ajj ,such as the proportion of the diagonal of a square to its side2 and that of the diameter of a circle1 to the periphery4 #aja ,the gnomons1 from which square numbers are produced1 are odd numbers in a natural series from unity1 i4 e4 #a1 #c1 #e1 #g1 #i1 #aa1 c4 for these1 added to each other continually1 produce square numbers ad infinitum4 ,but these gnomons continually decrease from the highest1 and are at length terminated by indivisible unity4 #ajb ,this doctrine of ineffable quantities1 or such whose proportion cannot be expressed1 is largely and accurately discussed by ,euclid1 in the tenth book of his ,elements3 but its study is neglected by modern mathematicians1 because it is of no use1 that is1 because it contributes to nothing mechanical4 #ajc ,this proposition is the #aath of the second book3 at least1 the method of dividing a line into extreme and mean proportion1 is immediately deduced from it2 which is done by ,euclid1 in the #cjth1 of the sixth book4 ,thus1 ,euclid shews 7#aa4 #b47 how to divide the line 7,a ,g ,b7diagram ,a ,b1 so that the rectangle under the whole ,a ,b1 and the segment ,g ,b1 may be equal to the square made from ,a ,g3 for when this is done1 it follows1 that as ,a ,b is to ,a ,g1 so is ,a ,g to ,g ,b2 as is well known4 ,but this proposition1 as ,dr4 ,barrow observes1 cannot be explained by numbers2 because there is not any number which can be so divided1 that the product from the whole into one part1 may be equal to the square from the other part4 #ajd ,all polygonous figures1 may1 it is well known1 be resolved into triangles2 and this is no less true of polygonous numbers1 as the following observations evince4 ,all number originates from indivisible unity1 which corresponds to a point3 and it is either linear1 corresponding to a line2 or superficial1 which corresponds to a superficies2 or solid1 which imitates a geometrical solid4 ,after unity1 therefore1 the first of linear numbers is the duad2 just as every finite line is allotted two extremities4 ,the triad is the first of superficial numbers2 as the triangle of geometrical figures4 ,and the tetrad1 is the first of solids2 because a triangular pyramid1 is the first among solid numbers1 as well as among solid figures4 ,as1 therefore1 the monad is assimilated to the point1 so the duad to the line1 the triad to the superficies1 and the tetrad to the solid4 ,now1 of superficial numbers1 some are triangles1 others squares1 others pentagons1 hexagons1 heptagons1 c4 ,triangular numbers are generated from the continual addition of numbers in a natural series1 beginning from unity4 ,thus1 if the numbers #a1 #b1 #c1 #d1 #e1 c4 be added to each other continually1 they will produce the triangular numbers #a1 #c1 #f1 #aj1 #ae1 c4 and if every triangular number be added to its preceding number1 it will produce a square number4 ,thus #c added to #a makes #d2 #f added to #c is equals #i2 #aj added to #f is equal to #af2 and so of the rest4 ,pentagons1 are produced from the junction of triangular and square numbers1 as follows4 ,let there be a series of triangular numbers #a1 #c1 #f1 #aj1 #ae1 c4 ,and of squares #a1 #d1 #i1 #af1 #be1 c4 ,then the second square number1 added to the first triangle1 will produce the first pentagon from unity1 i4e4 #e4 ,the third square added to the second triangle1 will produce the second pentagon1 i4e4 #ab2 and so of the rest1 by a similar addition4 ,in like manner1 the second pentagon1 added to the first triangle1 will form the first hexagon from unity2 the third pentagon and the second triangle1 will form the second hexagon1 c4 ,and1 by a similar proceeding1 all the other polygons may be obtained4 #aje ,intellections are universally correspondent to their objects1 and participate of evidence or the contrary1 in proportion as their subjects are lucid or obscure4 ,hence1 ,porphyry1 in his sentences1 justly observes1 that 8we do not understand in a similar manner with all the powers of the soul1 but according to the particular essence of each4 ,for with the intellect we understand intellectually2 and with the soul1 rationally3 our knowledge of plants is according to a seminal conception2 our understanding of bodies is imaginative2 and our intellection of the divinely solitary principle of the universe1 who is above all things1 is in a manner superior to intellectual perception1 and by a super-essential energy40 1 7#aj47 ,so that1 in consequence of this reasoning1 the speculations of geometry are then most true1 when most abstracted from sensible and material natures4 #ajf ,see ,plutarch1 in the life of ,marcellus4 #ajg ,in lib4 i4 de ,clo1 tex4 #bb4 et lib4 i4 ,meteo4 cap4 #c4 ,aristotle was called demoniacal by the ,platonic philosophers1 in consequence of the encomium bestowed on him by his master1 ,plato1 8,that he was the dmon of nature40 ,indeed1 his great knowledge in things subject to the dominion of nature1 well deserved this encomium2 and the epithet divine1 has been universally ascribed to ,plato1 from his profound knowledge of the intelligible world4 #ajh 1 is wanting in the original1 but is supplied by the excellent translation of ,barocius4 #aji 1 in the printed ,greek1 which ,fabricius1 in his ,bibliotheca ,grca1 vol4 i4 page #che1 is of opinion1 should be read 2 but ,i have rendered the word according to the translation of ,barocius1 who is likely to have obtained the true reading1 from the variety of manuscripts which he consulted4 #aaj ,the quadrature of the ,lunula is as follows4 ,illustration ,let ,a ,b ,c be a right-angled triangle1 and ,b ,a ,c a semi-circle on the diameter ,b ,c3 ,b ,n ,a a semi-circle described on the diameter ,a ,b2 ,a ,m ,c a semi-circle described on the diameter ,a ,c4 ,then the semi-circle ,b ,a ,c is equal to the semi-circle ,b ,n ,a1 and ,a ,m ,c together3 7because circles are to each other as the squares of their diameters1 #ca1 #f47 ,if1 therefore1 you take away the two spaces ,b ,a1 ,a ,c common on both sides1 there will remain the two lunulas ,b ,n ,a1 ,a ,m ,c1 bounded on both sides with circular lines1 equal to the right-angled triangle ,b ,a ,c4 ,and if the line ,b ,a1 be equal to the line ,a ,c1 and you let fall a perpendicular to the hypotenuse ,b ,c1 the triangle ,b ,a ,o will be equal to the lunular space ,b ,n ,a1 and the triangle ,c ,o ,a will be equal to the lunula ,c ,m ,a4 ,those who are curious1 may see a long account of an attempt of ,hippocrates to square the circle1 by the invention of the lunulas1 in ,simplicius on ,aristotle's ,physics1 lib4 i4 #aaa ,so ,barocius reads1 but ,fabricius 4 #aab i4 e4 ,the five regular bodies1 the pyramid1 cube1 octaedron1 dodecaedron and icosaedron2 concerning which1 and their application to the theory of the universe1 see ,kepler's admirable work1 ,de ,harmonia ,mundi4 #aac ,it may be doubted whether the optics and catoptrics1 ascribed to ,euclid in the editions of his works are genuine3 for ,savil1 and ,dr4 ,gregory1 think them scarcely worthy so great a man4 #aad ,there are two excellent editions of this work1 one by ,meibomius1 in his collection of ancient authors on harmony2 and the other by ,dr4 ,gregory1 in his collection of ,euclid's works4 #aae ,this work is most probably lost4 ,see ,dr4 ,gregory's ,euclid4 #aaf ,all this is shewn by ,proclus in the following ,commentaries2 and is surely most admirable and worthy the investigation of every liberal mind2 but ,i am afraid modern mathematicians very little regard such knowledge1 because it cannot be applied to practical and mechanical purposes4 #aag ,this work is unfortunately lost4 #aah ,because this is true only in isosceles and equilateral triangles4 #aai ,this follows from the #cb;d proposition of the first book of ,euclid2 and is demonstrated by ,dr4 ,barrow1 in his scholium to that proposition4 #abj ,the method of constructing these is shewn by our philosopher1 in his comment on the first proposition1 as will appear in the second volume of this work4 #aba ,the reader will please to observe1 that the definitions are1 indeed1 hypotheses1 according to the doctrine of ,plato1 as may be seen in the note to chap1 i4 book ,i4 of this work4 #abb ,in his last ,analytics4 ,see the preceding ,dissertation4 #abc ,that part of this work enclosed within the brackets1 is wanting in the original2 which ,i have restored from the excellent version of ,barocius4 ,the philosophical reader1 therefore1 of the original1 who may not have ,barocius in his possession1 will1 ,i hope1 be pleased1 to see so great a vacancy supplied2 especially1 as it contains the beginning of the commentary on the definition of a point4 #abd ,i do not find this nigma among the ,pythagoric symbols which are extant2 so that it is probably no where mentioned but in the present work4 ,and ,i am sorry to add1 that a figure and three oboli1 in too much the general cry of the present times4 #abe ,the present ,comment1 and indeed most of the following1 eminently evinces the truth of ,kepler's observation1 in his excellent work1 ,de ,harmonia ,mundi1 p4 #aah4 ,for1 speaking of our author's composition in the present work1 which he every where admires and defends1 he remarks as follows1 8oratio fluit ipsi torrentis instar1 ripas inundans1 et cca dubitationum vada gurgitesque occultans1 dum mens plena majestatis tantarum rerum1 luctatur in angustiis lingu1 et conclusio nunquam sibi ipsi verborum copi satisfaciens1 propositionum simplicitatem excedit40 ,but ,kepler was skilled in the ,platonic philosophy1 and appears to have been no less acquainted with the great depth of our author's mind than with the magnificence and sublimity of his language4 ,perhaps ,kepler is the only instance among the moderns1 of the philosophical and mathematical genius being united in the same person4 #abf ,that is1 the reason of a triangular figure 7for instance7 in the phantasy1 or triangle itself1 is superior to the triangular nature participated in that figure4 #abg ,in the tenth book of his ,republic4 #abh ,see the ,hymn to the ,mother of the ,gods1 in my translation of the ,orphic ,initiations4 #abi ,the philosopher here seems to contradict what he asserts in the end of his comment on the #acth ,definition3 for there he asserts1 that the circle is a certain plane space4 ,perhaps he may be reconciled1 by considering1 that as the circle subsists most according to bound1 when we speculate its essence in this respect we may define it according to the circumference1 which is the cause of its bound4 ,but when we consider it as participating of infinity also1 though not in so eminent a degree1 and view it from its emanations from the centre as well as in its regressions1 we may define it a plane space4 #acj ,that is1 the essential one of the soul is the mother of number2 but that which subsists in opinion is nothing more than the receptacle of the former2 just as matter is the seat of all forms4 ,for a farther account of the subsistence of numbers1 see the first section of the preceding ,dissertation4 #aca ,that is1 number composed from units4 #acb ,this sentence within the brackets1 is wholly omitted in the printed ,greek4 #acc ,in i4 ,de ,clo4 #acd ,this sentence within the brackets1 which is very imperfect in the ,greek1 ,i have supplied from the excellent translation of ,barocius4 ,in the ,greek there is nothing more than 4 #ace ,in the ,greek1 1 4 ,the latter part only of this oracle1 is to be found in all the printed editions of the ,zoroastrian oracles2 though it is wonderful how this omission could escape the notice of so may able critics1 and learned men4 ,it seems probable1 from hence1 that it is only to be found perfect in the present work4 #acf ,the word 1 is omitted in the ,greek4 #acg ,this and the following problems1 are the #ast1 #bb;d1 and #abth propositions of the first book4 ,but in the two last1 instead of the word or infinite1 which is the term employed by ,euclid1 ,mr4 ,simson1 in his edition of the ,elements1 uses the word unlimited4 ,but it is no unusual thing with this great geometrician1 to alter the words of ,euclid1 when they convey a philosophical meaning2 as we shall plainly evince in the course of these ,commentaries4 ,he certainly deserves the greatest praise for his zealous attachment to the ancient geometry3 but he would 7in my opinion7 have deserved still more1 had he been acquainted with the ,greek philosophy2 and fathomed the depth of ,proclus2 for then he would never have attempted to restore ,euclid's ,elements1 by depriving them of some very considerable beauties4 #ach ,this is doubtless the reason why the proportion between a right and circular line1 cannot be exactly obtained in numbers2 for on this hypothesis1 they must be incommensurable quantities2 because the one contains property essentially different from the other4 #aci ,illustration ,the cornicular angle is that which is made from the periphery of a circle and its tangent2 that is1 the angle comprehended by the arch ,l ,a1 and the right line ,f ,a1 which ,euclid in 7#af4 #c47 proves to be less than any right-lined angle4 ,and from this admirable proposition it follows1 by a legitimate consequence1 that any quantity may be continually and infinitely increased1 but another infinitely diminished2 and yet the augment of the first1 how great soever it may be1 shall always be less than the decrement of the second3 which ,cardan demonstrates as follows4 ,let there be proposed an angle of contact ,b ,a ,e1 and an acute angle ,h ,g ,i4 ,now if there be other lesser circles described ,a ,c1 ,a ,d1 the angle of contact will be evidently increased4 ,and if between the right lines ,g ,h1 ,g ,i1 there fall other right lines ,g ,k1 ,g ,l1 the acute angle shall be continually diminished3 yet the angle of contact1 however increased1 is always less than the acute angle1 however diminished4 ,sir ,isaac ,newton likewise observes1 in his ,treatise on ,fluxions1 that there are angles of contact made by other curve lines1 and their tangents infinitely less than those made by a circle and right line2 all which is demonstrably certain3 yet1 such is the force of prejudice1 that ,mr4 ,simson is of opinion1 with ,vieta1 that this part of the #afth proposition is adulterated2 and that the space made by a circular line and its tangent1 is no angle4 ,at least his words1 in the note upon this proposition1 will bear such a construction4 ,peletarius was likewise of the same opinion2 but is elaborately confuted by the excellent ,clavius1 as may be seen in his comment on this proposition4 ,but all the difficulties and paradoxes in this affair1 may be easily solved and admitted1 if we consider1 with our philosopher1 that the essence of an angle does not subsist in ether quantity1 quality1 or inclination1 taken singly1 but in the aggregate of them all4 ,for if we regard the inclination of a circular line to its tangent1 we shall find it possess the property1 by which ,euclid defines an angle3 if we respect its participation of quantity1 we shall find it capable of being augmented and diminished2 and if we regard it as possessing a peculiar quality1 we shall account for its being incommensurable with every right-lined angle4 ,see the ,comment on the #hth ,definition4 #adj ,in i4 ,de ,clo4 #ada ,it is from this cylindric spiral that the screw is formed4 #adb ,illustration ,the present very obscure passage1 may be explained by the following figure4 ,let ,a ,b ,c1 be a right angle1 and ,d ,e the line to be moved1 which is bisected in ,g4 ,now1 conceive it to be moved along the lines ,a ,b1 ,b ,c1 in such a manner1 that the point ,d may always remain in ,a ,b1 and the point ,e in ,b ,c4 ,then1 when the line ,d ,e1 is in the situations d e1 1 the point ,g1 shall be in g1 1 and these points ,g1 g1 1 shall be in a circle4 ,and any other point ,f in the line ,d ,e1 will1 at the same time1 describe an ellipsis2 the greater axis being in the line ,a ,b1 when the point ,f is between ,d and ,g2 and in the line ,b ,c1 when the point ,f is between ,g and ,e4 #adc ,that is1 the soul of the world4 #add ,in ,timo4 #ade ,the ellipsis4 #adf ,the cissoid4 ,for the properties of this curve1 see ,dr4 ,wallis's treatise on the cycloid1 p4 #ha4 #adg ,the conchoid4 #adh ,thus1 a right line1 when considered as the side of a parallelogram1 moving circularly1 generates a cylindrical superficies3 when moving circularly1 as the side of a triangle1 a conical surface2 and so in other lines1 the produced superficies varying according to the different positions of their generative lines4 #adi ,inv ii4 ,de ,rep4 #aej ,in multis locis4 #aea ,this definition is the same with that which ,mr4 ,simson has adopted instead of ,euclid's1 expressed in different words3 for he says1 8a plane superficies is that in which any two points being taken1 the straight line between them lies wholly in that superficies40 ,but he does not mention to whom he was indebted for the definition2 and this1 doubtless1 because he considered it was not worth while to relate the trifles of ,proclus at full length3 for these are his own words1 in his note to proposition #g1 book i4 ,nor has he informed us in what respect ,euclid's definition is indistinct4 #aeb ,in the ,greek 1 but it should doubtless be read 1 images1 as in the translation of ,barocius4 #aec ,mr4 ,simson1 in his note on this definition1 supposes it to be the addition of some less skilful editor2 on which account1 and because it is quite useless 7in his opinion7 he distinguishes it from the rest by inverted double commas4 ,but it is surely strange that the definition of angle in general should be accounted useless1 and the work of an unskilful geometrician4 ,such an assertion may1 indeed1 be very suitable to a professor of experimental philosophy1 who considers the useful as inseparable from practice2 but is by no means becoming a restorer of the liberal geometry of the ancients4 ,besides1 ,mr4 ,simson seems continually to forget that ,euclid was of the ,platonic sect2 and consequently was a philosopher as well as a mathematician4 ,i only add1 that the commentary on the present definition is1 in my opinion1 remarkably subtle and accurate1 and well deserves the profound attention of the greatest geometricians4 #aed ,for a philosophical discussion of the nature of quality and quantity1 consult the ,commentaries of ,ammonius1 and ,simplicius on ,aristotle's ,categories1 ,plotinus on the genera of beings1 and ,mr4 ,harris's ,philosophical ,arrangements4 #aee ,that is1 the ellipsis4 #aef ,that is1 they are either right1 acute1 or obtuse4 #aeg ,this oracle is not mentioned by any of the collectors of the ,zoroastrian oracles4 #aeh ,this1 indeed1 must always be the case with those geometricians1 who are not at the same time1 philosophers2 a conjunction no less valuable than rare4 ,hence1 from their ignorance of principles and intellectual concerns1 when any contemplative enquiry is proposed1 they immediately ask1 in what its utility consists2 considering every thing as superfluous1 which does not contribute to the solution of some practical problem4 #aei ,concerning the soul's descent into body1 see lib4 ix41 ,ennead iv4 of ,plotinus2 and for the method by which she may again return to her pristine felicity1 study the first book of ,porphyry's ,treatise on ,abstinence4 #afj ,this ,definition too1 is marked by ,mr4 ,simson with inverted commas1 as a symbol of its being interpolated4 ,but for what reason ,i know not1 unless because it is useless1 that is1 because it is philosophical6 #afa ,that is1 the various species of forms1 with which the four elements are replete4 #afb ,that is1 the circle4 #afc ,an admirer of the moderns1 and their pursuits1 will doubtless consider all this as the relics of heathen superstition and ignorance2 and will think1 perhaps1 he makes a great concession in admitting the existence of one supreme god1 without acknowledging a multitude of deities subordinate to the first4 ,for what the ancients can urge in defence of this obsolete opinion1 ,i must beg leave to refer the reader to the dissertation prefixed to my translation of ,orpheus2 in addition to which let him attend to the following considerations4 ,is it possible that the machinery of the gods in ,homer could be so beautiful1 if such beings had no existence8 ,or can any thing be beautiful which is destitute of all reality8 ,do not things universally please in proportion as they resemble reality8 ,perhaps it will be answered1 that the reverse of this is true1 and that fiction more generally pleases than truth1 as is evident from the great avidity with which romances are perused4 ,to this ,i reply1 that fiction itself ceases to be pleasing1 when it supposes absolute impossibilities3 for the existence of genii and fairies cannot be proved impossible2 and these compose all the marvellous of romance4 ,this observation is verified in ,spencer's ,fairy ,queen3 for his allegories1 in which the passions are personified1 are tedious and unpleasant1 because they are not disguised under the appearance of reality3 while the magic of ,circe1 the bower of ,calypso1 the rocks of ,scylla and ,charybdis1 and the melody of the ,syrens1 in the ,odyssey of ,homer1 though nothing but allegories1 universally enchant and delight1 because they are covered with the semblance of truth4 ,it is on this account that ,mikon's battles in heaven are barbarous and ridiculous in the extreme2 for every one sees the impossibility of supposing gun-powder and cannons in the celestial regions3 the machinery is forced and unnatural1 contains no elegance of fancy1 and is not replete with any mystical information4 ,on the contrary1 ,homer's machinery is natural and possible1 is full of dignity and elegance1 and is pregnant with the sublimest truths2 it delights and enobles the mind of the reader1 astonishes him with its magnificence and propriety1 and animates him with the fury of poetic inspiration4 ,and this1 because it is possible and true4 #afd ,the sentence within the brackets is omitted in the ,greek4 #afe ,that is1 the circular form proceeds from bound1 but right-lined figures from infinity4 #aff ,that is1 the number three4 #afg ,in ,timo4 #afh 1 or1 to which1 is wanting in the original1 and in all the published collections of the ,zoroastrian oracles4 #afi ,that is ,jupiter1 who is called triadic1 because he proceeds from ,saturn and ,rhea2 and because his government is participated by ,neptune and ,pluto1 for each of these is called ,jupiter by ,orpheus4 #agj ,this sentence1 within the brackets1 is omitted in the printed ,greek4 #aga ,illustration3 ,fig4 ,i4 ,illustration3 ,fig4 ,,ii4 ,thus let a part ,a ,e ,b cut off by the diameter ,a ,b 7fig4 ,i47 of the circle ,a ,e ,b ,d be placed on the other part ,a ,d ,b1 as in fig4 ,,ii44 ,then1 if it is not equal to the other part1 either ,a ,e ,b will fall within ,a ,d ,b1 or ,a ,d ,b within ,a ,e ,b3 but in either case1 ,c ,e will be equal to ,c ,d1 which is absurd4 #agb ,this objection is urged by ,philoponus1 in his book against ,proclus on the eternity of the world2 but not1 in my opinion1 with any success4 ,see also ,simplicius1 in his third digression against ,philoponus1 in his commentary on the #hth book of ,aristotle's ,physics4 #agc ,this definition is no where extant but in the commentaries of ,proclus4 ,instead of it1 in almost all the printed editions of ,euclid1 the following is substituted4 ,a segment of a circle is the figure contained by a diameter1 and the part of the circumference cut off by the diameter4 ,this ,mr4 ,simson has marked with commas1 as a symbol of its being interpolated3 but he has taken no notice of the different reading in the commentaries of ,proclus4 ,and what is still more remarkable1 this variation is not noticed by any editor of ,euclid's ,elements1 either ancient or modern4 #agd ,as in every hyperbola4 #age ,the ,platonic reader must doubtless be pleased to find that ,euclid was deeply skilled in the philosophy of ,plato1 as ,proclus every where evinces4 ,indeed1 the great accuracy1 and elegant distribution of these ,elements1 sufficiently prove the truth of this assertion4 ,and it is no inconsiderable testimony in favour of the ,platonic philosophy1 that its assistance enabled ,euclid to produce such an admirable work4 #agf ,concerning these crowns1 or annular spaces1 consult the great work of that very subtle and elegant mathematician ,tacquet1 entitled ,cylindrica et ,annularia4 #agg ,in the preceding tenth commentary4 #agh ,this in consequence of every triangle possessing angles alone equal to two right4 #agi ,this too1 follows from the same cause as above4 #ahj ,thus the following figure ,a ,b ,d ,c has four sides1 and but three angles4 ,illustration #aha ,the ,greek in this place is very erroneous1 which ,i have restored from the version of ,barocius4 #ahb ,for the ,greek word is derived from the verb 1 which signifies to have a circumvolute motion4 #ahc ,see the ,orphic ,hymns of ,onomacritus to these deities2 my translation of which ,i must recommend to the ,english reader1 because there is no other4 #ahd ,these twelve divinities1 of which ,jupiter is the head1 are1 ,jupiter1 ,neptune1 ,vulcan1 ,vesta1 ,minerva1 ,mars1 ,ceres1 ,juno1 ,diana1 ,mercury1 ,venus1 and ,apollo4 ,the first triad of these is demiurgic1 the second comprehends guardian deities1 the third is vivific1 or zoogonic1 and the fourth contains elevating gods4 ,but1 for a particular theological account of these divinities1 study ,proclus on ,plato's ,theology1 and you will find their nature unfolded1 in page #djc1 of that admirable work4 #ahe ,for it is easy to conceive a cylindric spiral described about a right-line1 so as to preserve an equal distance from it in every part2 and in this case the spiral and right-line will never coincide though infinitely produced4 #ahf ,illustration ,as the conchoid is a curve but little known1 ,i have subjoined the following account of its generation and principal property4 ,in any given right line ,a ,p1 call ,p the pole1 ,a the vertex1 and any intermediate point ,c the centre of the conchoid3 likewise1 conceive an infinite right line ,c ,h1 which is called a rule1 perpendicular to ,a ,p4 ,then1 if the right line ,a p continued at p as much as is necessary1 is conceived to be so turned about the abiding pole p1 that the point ,c may perpetually remain in the right line ,c ,h1 the point ,a will describe the curve ,a o1 which the ancients called a conchoid4 ,in this curve it is manifest 7on account of the right line ,p ,o1 cutting the rule in ,h that the point o will never arrive at rule ,c ,h2 but because h ,o is perpetually equal to ,c ,a1 and the angle of section is continually more acute1 the distance of the point ,o from ,c ,h will at length be less than any given distance1 and consequently the right line ,c ,h will be an asymptote to the curve ,a ,o4 ,when the pole is at ,p1 so that ,p ,c is equal to ,c ,a1 the conchoid ,a ,o described by the revolution of ,p ,a1 is called a primary conchoid1 and those described from the poles p1 and 1 or the curves ,a o1 ,a 1 secondary conchoids2 and these are either contracted or protracted1 as the eccentricity ,p ,c1 is greater or less than the generative radius ,c ,a1 which is called the altitude of the curve4 ,now1 from the nature of the conchoid1 it may be easily inferred1 that not only the exterior conchoid ,a will never coincide with the right line ,c ,h1 but this is likewise true of the conchoids ,a ,o1 ,a o2 and by infinitely extending the right-line ,a 1 an infinite number of conchoids may be described between the exterior conchoid ,a 1 and the line ,c ,h1 no one of which shall ever coincide with the asymptote ,c ,h4 ,and this paradoxical property of the conchoid which has not been observed by any mathematician1 is a legitimate consequence of the infinite divisibility of quantity4 ,not1 indeed1 that quantity admits of an actual division in infinitum1 for this is absurd and impossible2 but it is endued with an unwearied capacity of division1 and a power of being diffused into multitude1 which can never be exhausted4 ,and this infinite capacity which it possesses arises from its participation of the indefinite duad2 the source of boundless diffusion1 and innumerable multitude4 ,illustration ,but this singular property is not confined to the conchoid1 but is found in the following curve4 ,conceive that the right line ,a ,c which is perpendicular to the indefinite line ,x ,y1 is equal to the quadrantal arch ,h ,d1 described from the centre ,c1 with the radius ,c ,d3 then from the same centre ,c1 with the several distances ,c ,e1 ,c ,f1 ,c ,g1 describe the arches ,e l1 ,f n1 ,g p1 each of which must be conceived equal to the first arch ,h ,d1 and so on infinitely4 ,now1 if the points ,h1 k1 l1 n1 p1 be joined1 they will form a curve line1 approaching continually nearer to the right-line ,a ,b 7parallel to ,c ,y7 but never effecting a perfect coincidence4 ,this will be evident from considering that each of the sines of the arches ,h ,d1 l ,e1 n ,f1 c4 being less than its respective arch1 must also be less than the right-line ,a ,c1 and consequently can never coincide with the right-line ,a ,b4 ,but if other arches ,d i1 ,e m1 ,f o1 c4 each of them equal to the right-line ,a ,c1 and described from one centre1 tangents to the former arches ,h ,d1 l ,e1 n ,f1 c4 be supposed2 it is evident that the points ,h1 i1 m1 o1 c4 being joined1 will form a curve line1 which shall pass beyond the former curve1 and converge still nearer to the line ,a ,b1 without a possibility of ever becoming coincident3 for since the arches ,d i1 ,e m1 ,f o1 c4 have less curvature than the former arches1 but are equal to them in length1 it is evident that they will be subtended by longer lines1 and yet can never touch the right-line ,a ,b4 ,in like manner1 if other tangent arches be drawn to the former1 and so on infinitely1 with the same conditions1 an infinite number of curve-lines will be formed1 each of them passing between ,h p and ,a ,b1 and continually diverging from the latter1 without a possibility of ever coinciding with the former4 ,this curve1 which ,i invented some years since1 ,i suspect to be a parabola2 but ,i have not yet had opportunity to determine it with certainty4 ,transcriber's ,notes3 #a4 ,obvious printers'1 punctuation and spelling errors have been corrected silently4 #b4 ,where hyphenation is in doubt1 it has been retained as in the original4 #c4 ,some hyphenated and non-hyphenated versions of the same words have been retained as in the original4 #d4 ,the errata have been soilently corrected4 #e4 ,italics are shown as xxx4