Chapter 56

[128]See the Hymn to the Mother of the Gods, in my translation of the Orphic Initiations.[129]The philosopher here seems to contradict what he asserts in the end of his comment on the 13th Definition: for there he asserts, that the circle is a certain plane space. Perhaps he may be reconciled, by considering, that as the circle subsists most according to bound, when we speculate its essence in this respect we may define it according to the circumference, which is the cause of its bound. But when we consider it as participating of infinity also, though not in so eminent a degree, and view it from its emanations from the centre as well as in its regressions, we may define it a plane space.[130]That is, the essential one of the soul is the mother of number; but that which subsists in opinion is nothing more than the receptacle of the former; just as matter is the seat of all forms. For a farther account of the subsistence of numbers, see the first section of the preceding Dissertation.[131]That is, number composed from units.[132]This sentence within the brackets, is wholly omitted in the printed Greek.[133]In i. De Cælo.[134]This sentence within the brackets, which is very imperfect in the Greek, I have supplied from the excellent translation of Barocius. In the Greek there is nothing more than λὲγω δὲ ἑνὸν τῂν γραμμὴν δυαδός πρὸς τὸ στερεόν.[135]In the Greek, γὰρ ἡ μονὰς ἐκεῖ πρῶτον, ὅπου πατρικὴ μονάς ἐστι φησὶ τὸ λόγιον. The latter part only of this oracle, is to be found in all the printed editions of the Zoroastrian oracles; though it is wonderful how this omission could escape the notice of so may able critics, and learned men. It seems probable, from hence, that it is only to be found perfect in the present work.[136]The word τανάη, is omitted in the Greek.[137]This and the following problems, are the 1st, 22d, and 12th propositions of the first book. But in the two last, instead of the word ἄπειρος or infinite, which is the term employed by Euclid, Mr. Simson, in his edition of the Elements, uses the word unlimited. But it is no unusual thing with this great geometrician, to alter the words of Euclid, when they convey a philosophical meaning; as we shall plainly evince in the course of these Commentaries. He certainly deserves the greatest praise for his zealous attachment to the ancient geometry: but he would (in my opinion) have deserved still more, had he been acquainted with the Greek philosophy; and fathomed the depth of Proclus; for then he would never have attempted to restore Euclid’s Elements, by depriving them of some very considerable beauties.[138]This is doubtless the reason why the proportion between a right and circular line, cannot be exactly obtained in numbers; for on this hypothesis, they must be incommensurable quantities; because the one contains property essentially different from the other.[139]The cornicular angle is that which is made from the periphery of a circle and its tangent; that is, the angle comprehended by the arch L A, and the right line F A, which Euclid in (16. 3.) proves to be less than any right-lined angle. And from this admirable proposition it follows, by a legitimate consequence, that any quantity may be continually and infinitely increased, but another infinitely diminished; and yet the augment of the first, how great soever it may be, shall always be less than the decrement of the second: which Cardan demonstrates as follows. Let there be proposed an angle of contact B A E, and an acute angle H G I. Now if there be other lesser circles described A C, A D, the angle of contact will be evidently increased. And if between the right lines G H, G I, there fall other right lines G K, G L, the acute angle shall be continually diminished: yet the angle of contact, however increased, is always less than the acute angle, however diminished. Sir Isaac Newton likewise observes, in his Treatise on Fluxions, that there are angles of contact made by other curve lines, and their tangents infinitely less than those made by a circle and right line; all which is demonstrably certain: yet, such is the force of prejudice, that Mr. Simson is of opinion, with Vieta, that this part of the 16th proposition is adulterated; and that the space made by a circular line and its tangent, is no angle. At least his words, in the note upon this proposition, will bear such a construction. Peletarius was likewise of the same opinion; but is elaborately confuted by the excellent Clavius, as may be seen in his comment on this proposition. But all the difficulties and paradoxes in this affair, may be easily solved and admitted, if we consider, with our philosopher, that the essence of an angle does not subsist in ether quantity, quality, or inclination, taken singly, but in the aggregate of them all. For if we regard the inclination of a circular line to its tangent, we shall find it possess the property, by which Euclid defines an angle: if we respect its participation of quantity, we shall find it capable of being augmented and diminished; and if we regard it as possessing a peculiar quality, we shall account for its being incommensurable with every right-lined angle. See the Comment on the 8th Definition.[140]In i. De Cælo.[141]It is from this cylindric spiral that the screw is formed.[142]The present very obscure passage, may be explained by the following figure. Let A B C, be a right angle, and D E the line to be moved, which is bisected in G. Now, conceive it to be moved along the lines A B, B C, in such a manner, that the point D may always remain in A B, and the point E in B C. Then, when the line D E, is in the situationsd e,δ ε, the point G, shall be ing, γ, and these points G,g, γ, shall be in a circle. And any other point F in the line D E, will, at the same time, describe an ellipsis; the greater axis being in the line A B, when the point F is between D and G; and in the line B C, when the point F is between G and E.[143]That is, the soul of the world.[144]In Timæo.[145]The ellipsis.[146]The cissoid. For the properties of this curve, see Dr. Wallis’s treatise on the cycloid, p. 81.[147]The conchoid.[148]Thus, a right line, when considered as the side of a parallelogram, moving circularly, generates a cylindrical superficies: when moving circularly, as the side of a triangle, a conical surface; and so in other lines, the produced superficies varying according to the different positions of their generative lines.[149]Inv ii. De Rep.[150]In multis locis.[151]This definition is the same with that which Mr. Simson has adopted instead of Euclid’s, expressed in different words: for he says, “a plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.” But he does not mention to whom he was indebted for the definition; and this, doubtless, because he consideredit was not worth while to relate the trifles of Proclus at full length: for these are his own words, in his note to proposition 7, book i. Nor has he informed us in what respect Euclid’s definition isindistinct.[152]In the Greek ἐννοιὰς, but it should doubtless be read εἰκόνας,images, as in the translation of Barocius.[153]Mr. Simson, in his note on this definition, supposes it to be the addition of some less skilful editor; on which account, and because it is quite useless (in his opinion) he distinguishes it from the rest by inverted double commas. But it is surely strange that the definition of angle in general should be accounted useless, and the work of an unskilful geometrician. Such an assertion may, indeed, be very suitable to a professor of experimental philosophy, who considers the useful as inseparable from practice; but is by no means becoming a restorer of the liberal geometry of the ancients. Besides, Mr. Simson seems continually to forget that Euclid was of the Platonic sect; and consequently was a philosopher as well as a mathematician. I only add, that the commentary on the present definition is, in my opinion, remarkably subtle and accurate, and well deserves the profound attention of the greatest geometricians.[154]For a philosophical discussion of the nature of quality and quantity, consult the Commentaries of Ammonius, and Simplicius on Aristotle’s Categories, Plotinus on the genera of beings, and Mr. Harris’s Philosophical Arrangements.[155]That is, the ellipsis.[156]That is, they are either right, acute, or obtuse.[157]This oracle is not mentioned by any of the collectors of the Zoroastrian oracles.[158]This, indeed, must always be the case with those geometricians, who are not at the same time, philosophers; a conjunction no less valuable than rare. Hence, from their ignorance of principles and intellectual concerns, when any contemplative enquiry is proposed, they immediately ask, in what its utility consists; considering every thing as superfluous, which does not contribute to the solution of some practical problem.[159]Concerning the soul’s descent into body, see lib. ix., Ennead iv. of Plotinus; and for the method by which she may again return to her pristine felicity, study the first book of Porphyry’s Treatise on Abstinence.[160]This Definition too, is marked by Mr. Simson with inverted commas, as a symbol of its being interpolated. But for what reason I know not, unless because it is useless, that is, because it isphilosophical![161]That is, the various species of forms, with which the four elements are replete.[162]That is, the circle.[163]An admirer of the moderns, and their pursuits, will doubtless consider all this as the relics of heathen superstition and ignorance; and will think, perhaps, he makes a great concession in admitting the existence of one supreme god, without acknowledging a multitude of deities subordinate to the first. For what the ancients can urge in defence of this obsolete opinion, I must beg leave to refer the reader to the dissertation prefixed to my translation of Orpheus; in addition to which let him attend to the following considerations. Is it possible that the machinery of the gods in Homer could be so beautiful, if such beings had no existence? Or can any thing be beautiful which is destitute of all reality? Do not things universally please in proportion as they resemble reality? Perhaps it will be answered, that the reverse of this is true, and that fiction more generally pleases than truth, as is evident from the great avidity with which romances are perused. To this I reply, that fiction itself ceases to be pleasing, when it supposes absolute impossibilities: for the existence of genii and fairies cannot be proved impossible; and these compose all the marvellous of romance. This observation is verified in Spencer’s Fairy Queen: for his allegories, in which the passions are personified, are tedious and unpleasant, because they are not disguised under the appearance of reality: while the magic of Circe, the bower of Calypso, the rocks of Scylla and Charybdis, and the melody of the Syrens, in the Odyssey of Homer, though nothing but allegories, universally enchant and delight, because they are covered with the semblance of truth. It is on this account that Mikon’s battles in heaven are barbarous and ridiculous in the extreme; for every one sees the impossibility of supposing gun-powder and cannons in the celestial regions: the machinery is forced and unnatural, contains no elegance of fancy, and is not replete with any mystical information. On the contrary, Homer’s machinery is natural and possible, is full of dignity and elegance, and is pregnant with the sublimest truths; it delights and enobles the mind of the reader, astonishes him with its magnificence and propriety, and animates him with the fury of poetic inspiration. And this, because it is possible and true.[164]The sentence within the brackets is omitted in the Greek.[165]That is, the circular form proceeds frombound, but right-lined figures frominfinity.[166]That is, the number three.[167]In Timæo.[168]πρὸς ὃ, or,to which, is wanting in the original, and in all the published collections of the Zoroastrian oracles.[169]That is Jupiter, who is called triadic, because he proceeds from Saturn and Rhea; and because his government is participated by Neptune and Pluto, for each of these is called Jupiter by Orpheus.[170]This sentence, within the brackets, is omitted in the printed Greek.[171]Fig. I. Fig. II.Thus let a part A E B cut off by the diameter A B (fig. I.) of the circle A E B D be placed on the other part A D B, as in fig. II. Then, if it is not equal to the other part, either A E B will fall within A D B, or A D B within A E B: but in either case, C E will be equal to C D, which is absurd.[172]This objection is urged by Philoponus, in his book against Proclus on the eternity of the world; but not, in my opinion, with any success. See also Simplicius, in his third digression against Philoponus, in his commentary on the 8th book of Aristotle’s Physics.[173]This definition is no where extant but in the commentaries of Proclus. Instead of it, in almost all the printed editions of Euclid, the following is substituted.A segment of a circle is the figure contained by a diameter, and the part of the circumference cut off by the diameter.This Mr. Simson has marked with commas, as a symbol of its being interpolated: but he has taken no notice of the different reading in the commentaries of Proclus. And what is still more remarkable, this variation is not noticed by any editor of Euclid’s Elements, either ancient or modern.[174]As in every hyperbola.[175]The Platonic reader must doubtless be pleased to find that Euclid was deeply skilled in the philosophy of Plato, as Proclus every where evinces. Indeed, the great accuracy, and elegant distribution of these Elements, sufficiently prove the truth of this assertion. And it is no inconsiderable testimony in favour of the Platonic philosophy, that its assistance enabled Euclid to produce such an admirable work.[176]Concerning these crowns, or annular spaces, consult the great work of that very subtle and elegant mathematician Tacquet, entitledCylindrica et Annularia.[177]In the preceding tenth commentary.[178]This in consequence of every triangle possessing angles alone equal to two right.[179]This too, follows from the same cause as above.[180]Thus the following figure A B D C has four sides, and but three angles.[181]The Greek in this place is very erroneous, which I have restored from the version of Barocius.[182]For the Greek word ῥόμβος is derived from the verb ῥέμβω, which signifies to have a circumvolute motion.[183]See the Orphic Hymns of Onomacritus to these deities; my translation of which I must recommend to the English reader, because there is no other.[184]These twelve divinities, of which Jupiter is the head, are,Jupiter,Neptune,Vulcan,Vesta,Minerva,Mars,Ceres,Juno,Diana,Mercury,Venus, andApollo. The first triad of these is demiurgic, the second comprehends guardian deities, the third is vivific, or zoogonic, and the fourth contains elevating gods. But, for a particular theological account of these divinities, study Proclus on Plato’s Theology, and you will find their nature unfolded, in page 403, of that admirable work.[185]For it is easy to conceive a cylindric spiral described about a right-line, so as to preserve an equal distance from it in every part; and in this case the spiral and right-line will never coincide though infinitely produced.[186]As the conchoid is a curve but little known, I have subjoined the following account of its generation and principal property. In any given right line A P, call P the pole, A the vertex, and any intermediate point C the centre of the conchoid: likewise, conceive an infinite right line C H, which is called a rule, perpendicular to A P. Then, if the right line Apcontinued atpas much as is necessary, is conceived to be so turned about the abiding polep, that the point C may perpetually remain in the right line C H, the point A will describe the curve Ao, which the ancients called a conchoid.In this curve it is manifest (on account of the right line P O, cutting the rule in H that the pointowill never arrive at rule C H; but becausehO is perpetually equal to C A, and the angle of section is continually more acute, the distance of the point O from C H will at length be less than any given distance, and consequently the right line C H will be an asymptote to the curve A O.When the pole is at P, so that P C is equal to C A, the conchoid A O described by the revolution of P A, is called a primary conchoid, and those described from the polesp, andπ, or the curves Ao, Aω, secondary conchoids; and these are either contracted or protracted, as the eccentricity P C, is greater or less than the generative radius C A, which is called the altitude of the curve.Now, from the nature of the conchoid, it may be easily inferred, that not only the exterior conchoid Aωwill never coincide with the right line C H, but this is likewise true of the conchoids A O, Ao; and by infinitely extending the right-line Aπ, an infinite number of conchoids may be described between the exterior conchoid Aω, and the line C H, no one of which shall ever coincide with the asymptote C H. And this paradoxical property of the conchoid which has not been observed by any mathematician, is a legitimate consequence of the infinite divisibility of quantity. Not, indeed, that quantity admits of an actual division in infinitum, for this is absurd and impossible; but it is endued with an unwearied capacity of division, and a power of being diffused into multitude, which can never be exhausted. And this infinite capacity which it possesses arises from its participation of the indefinite duad; the source of boundless diffusion, and innumerable multitude.But this singular property is not confined to the conchoid, but is found in the following curve. Conceive that the right line A C which is perpendicular to the indefinite line X Y, is equal to the quadrantal arch H D, described from the centre C, with the radius C D: then from the same centre C, with the several distances C E, C F, C G, describe the arches El, Fn, Gp, each of which must be conceived equal to the first arch H D, and so on infinitely. Now, if the points H,k,l,n,p, be joined, they will form a curve line, approaching continually nearer to the right-line A B (parallel to C Y) but never effecting a perfect coincidence. This will be evident from considering that each of the sines of the arches H D,lE,nF, &c. being less than its respective arch, must also be less than the right-line A C, and consequently can never coincide with the right-line A B.But if other arches Di, Em, Fo, &c. each of them equal to the right-line A C, and described from one centre, tangents to the former arches H D,lE,nF, &c. be supposed; it is evident that the points H,i,m,o, &c. being joined, will form a curve line, which shall pass beyond the former curve, and converge still nearer to the line A B, without a possibility of ever becoming coincident: for since the arches Di, Em, Fo, &c. have less curvature than the former arches, but are equal to them in length, it is evident that they will be subtended by longer lines, and yet can never touch the right-line A B. In like manner, if other tangent arches be drawn to the former, and so on infinitely, with the same conditions, an infinite number of curve-lines will be formed, each of them passing between Hpand A B, and continually diverging from the latter, without a possibility of ever coinciding with the former. This curve, which I invented some years since, I suspect to be a parabola; but I have not yet had opportunity to determine it with certainty.

[128]See the Hymn to the Mother of the Gods, in my translation of the Orphic Initiations.

[129]The philosopher here seems to contradict what he asserts in the end of his comment on the 13th Definition: for there he asserts, that the circle is a certain plane space. Perhaps he may be reconciled, by considering, that as the circle subsists most according to bound, when we speculate its essence in this respect we may define it according to the circumference, which is the cause of its bound. But when we consider it as participating of infinity also, though not in so eminent a degree, and view it from its emanations from the centre as well as in its regressions, we may define it a plane space.

[130]That is, the essential one of the soul is the mother of number; but that which subsists in opinion is nothing more than the receptacle of the former; just as matter is the seat of all forms. For a farther account of the subsistence of numbers, see the first section of the preceding Dissertation.

[131]That is, number composed from units.

[132]This sentence within the brackets, is wholly omitted in the printed Greek.

[133]In i. De Cælo.

[134]This sentence within the brackets, which is very imperfect in the Greek, I have supplied from the excellent translation of Barocius. In the Greek there is nothing more than λὲγω δὲ ἑνὸν τῂν γραμμὴν δυαδός πρὸς τὸ στερεόν.

[135]In the Greek, γὰρ ἡ μονὰς ἐκεῖ πρῶτον, ὅπου πατρικὴ μονάς ἐστι φησὶ τὸ λόγιον. The latter part only of this oracle, is to be found in all the printed editions of the Zoroastrian oracles; though it is wonderful how this omission could escape the notice of so may able critics, and learned men. It seems probable, from hence, that it is only to be found perfect in the present work.

[136]The word τανάη, is omitted in the Greek.

[137]This and the following problems, are the 1st, 22d, and 12th propositions of the first book. But in the two last, instead of the word ἄπειρος or infinite, which is the term employed by Euclid, Mr. Simson, in his edition of the Elements, uses the word unlimited. But it is no unusual thing with this great geometrician, to alter the words of Euclid, when they convey a philosophical meaning; as we shall plainly evince in the course of these Commentaries. He certainly deserves the greatest praise for his zealous attachment to the ancient geometry: but he would (in my opinion) have deserved still more, had he been acquainted with the Greek philosophy; and fathomed the depth of Proclus; for then he would never have attempted to restore Euclid’s Elements, by depriving them of some very considerable beauties.

[138]This is doubtless the reason why the proportion between a right and circular line, cannot be exactly obtained in numbers; for on this hypothesis, they must be incommensurable quantities; because the one contains property essentially different from the other.

[139]The cornicular angle is that which is made from the periphery of a circle and its tangent; that is, the angle comprehended by the arch L A, and the right line F A, which Euclid in (16. 3.) proves to be less than any right-lined angle. And from this admirable proposition it follows, by a legitimate consequence, that any quantity may be continually and infinitely increased, but another infinitely diminished; and yet the augment of the first, how great soever it may be, shall always be less than the decrement of the second: which Cardan demonstrates as follows. Let there be proposed an angle of contact B A E, and an acute angle H G I. Now if there be other lesser circles described A C, A D, the angle of contact will be evidently increased. And if between the right lines G H, G I, there fall other right lines G K, G L, the acute angle shall be continually diminished: yet the angle of contact, however increased, is always less than the acute angle, however diminished. Sir Isaac Newton likewise observes, in his Treatise on Fluxions, that there are angles of contact made by other curve lines, and their tangents infinitely less than those made by a circle and right line; all which is demonstrably certain: yet, such is the force of prejudice, that Mr. Simson is of opinion, with Vieta, that this part of the 16th proposition is adulterated; and that the space made by a circular line and its tangent, is no angle. At least his words, in the note upon this proposition, will bear such a construction. Peletarius was likewise of the same opinion; but is elaborately confuted by the excellent Clavius, as may be seen in his comment on this proposition. But all the difficulties and paradoxes in this affair, may be easily solved and admitted, if we consider, with our philosopher, that the essence of an angle does not subsist in ether quantity, quality, or inclination, taken singly, but in the aggregate of them all. For if we regard the inclination of a circular line to its tangent, we shall find it possess the property, by which Euclid defines an angle: if we respect its participation of quantity, we shall find it capable of being augmented and diminished; and if we regard it as possessing a peculiar quality, we shall account for its being incommensurable with every right-lined angle. See the Comment on the 8th Definition.

[139]

The cornicular angle is that which is made from the periphery of a circle and its tangent; that is, the angle comprehended by the arch L A, and the right line F A, which Euclid in (16. 3.) proves to be less than any right-lined angle. And from this admirable proposition it follows, by a legitimate consequence, that any quantity may be continually and infinitely increased, but another infinitely diminished; and yet the augment of the first, how great soever it may be, shall always be less than the decrement of the second: which Cardan demonstrates as follows. Let there be proposed an angle of contact B A E, and an acute angle H G I. Now if there be other lesser circles described A C, A D, the angle of contact will be evidently increased. And if between the right lines G H, G I, there fall other right lines G K, G L, the acute angle shall be continually diminished: yet the angle of contact, however increased, is always less than the acute angle, however diminished. Sir Isaac Newton likewise observes, in his Treatise on Fluxions, that there are angles of contact made by other curve lines, and their tangents infinitely less than those made by a circle and right line; all which is demonstrably certain: yet, such is the force of prejudice, that Mr. Simson is of opinion, with Vieta, that this part of the 16th proposition is adulterated; and that the space made by a circular line and its tangent, is no angle. At least his words, in the note upon this proposition, will bear such a construction. Peletarius was likewise of the same opinion; but is elaborately confuted by the excellent Clavius, as may be seen in his comment on this proposition. But all the difficulties and paradoxes in this affair, may be easily solved and admitted, if we consider, with our philosopher, that the essence of an angle does not subsist in ether quantity, quality, or inclination, taken singly, but in the aggregate of them all. For if we regard the inclination of a circular line to its tangent, we shall find it possess the property, by which Euclid defines an angle: if we respect its participation of quantity, we shall find it capable of being augmented and diminished; and if we regard it as possessing a peculiar quality, we shall account for its being incommensurable with every right-lined angle. See the Comment on the 8th Definition.

[140]In i. De Cælo.

[141]It is from this cylindric spiral that the screw is formed.

[142]The present very obscure passage, may be explained by the following figure. Let A B C, be a right angle, and D E the line to be moved, which is bisected in G. Now, conceive it to be moved along the lines A B, B C, in such a manner, that the point D may always remain in A B, and the point E in B C. Then, when the line D E, is in the situationsd e,δ ε, the point G, shall be ing, γ, and these points G,g, γ, shall be in a circle. And any other point F in the line D E, will, at the same time, describe an ellipsis; the greater axis being in the line A B, when the point F is between D and G; and in the line B C, when the point F is between G and E.

[142]

The present very obscure passage, may be explained by the following figure. Let A B C, be a right angle, and D E the line to be moved, which is bisected in G. Now, conceive it to be moved along the lines A B, B C, in such a manner, that the point D may always remain in A B, and the point E in B C. Then, when the line D E, is in the situationsd e,δ ε, the point G, shall be ing, γ, and these points G,g, γ, shall be in a circle. And any other point F in the line D E, will, at the same time, describe an ellipsis; the greater axis being in the line A B, when the point F is between D and G; and in the line B C, when the point F is between G and E.

[143]That is, the soul of the world.

[144]In Timæo.

[145]The ellipsis.

[146]The cissoid. For the properties of this curve, see Dr. Wallis’s treatise on the cycloid, p. 81.

[147]The conchoid.

[148]Thus, a right line, when considered as the side of a parallelogram, moving circularly, generates a cylindrical superficies: when moving circularly, as the side of a triangle, a conical surface; and so in other lines, the produced superficies varying according to the different positions of their generative lines.

[149]Inv ii. De Rep.

[150]In multis locis.

[151]This definition is the same with that which Mr. Simson has adopted instead of Euclid’s, expressed in different words: for he says, “a plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.” But he does not mention to whom he was indebted for the definition; and this, doubtless, because he consideredit was not worth while to relate the trifles of Proclus at full length: for these are his own words, in his note to proposition 7, book i. Nor has he informed us in what respect Euclid’s definition isindistinct.

[152]In the Greek ἐννοιὰς, but it should doubtless be read εἰκόνας,images, as in the translation of Barocius.

[153]Mr. Simson, in his note on this definition, supposes it to be the addition of some less skilful editor; on which account, and because it is quite useless (in his opinion) he distinguishes it from the rest by inverted double commas. But it is surely strange that the definition of angle in general should be accounted useless, and the work of an unskilful geometrician. Such an assertion may, indeed, be very suitable to a professor of experimental philosophy, who considers the useful as inseparable from practice; but is by no means becoming a restorer of the liberal geometry of the ancients. Besides, Mr. Simson seems continually to forget that Euclid was of the Platonic sect; and consequently was a philosopher as well as a mathematician. I only add, that the commentary on the present definition is, in my opinion, remarkably subtle and accurate, and well deserves the profound attention of the greatest geometricians.

[154]For a philosophical discussion of the nature of quality and quantity, consult the Commentaries of Ammonius, and Simplicius on Aristotle’s Categories, Plotinus on the genera of beings, and Mr. Harris’s Philosophical Arrangements.

[155]That is, the ellipsis.

[156]That is, they are either right, acute, or obtuse.

[157]This oracle is not mentioned by any of the collectors of the Zoroastrian oracles.

[158]This, indeed, must always be the case with those geometricians, who are not at the same time, philosophers; a conjunction no less valuable than rare. Hence, from their ignorance of principles and intellectual concerns, when any contemplative enquiry is proposed, they immediately ask, in what its utility consists; considering every thing as superfluous, which does not contribute to the solution of some practical problem.

[159]Concerning the soul’s descent into body, see lib. ix., Ennead iv. of Plotinus; and for the method by which she may again return to her pristine felicity, study the first book of Porphyry’s Treatise on Abstinence.

[160]This Definition too, is marked by Mr. Simson with inverted commas, as a symbol of its being interpolated. But for what reason I know not, unless because it is useless, that is, because it isphilosophical!

[161]That is, the various species of forms, with which the four elements are replete.

[162]That is, the circle.

[163]An admirer of the moderns, and their pursuits, will doubtless consider all this as the relics of heathen superstition and ignorance; and will think, perhaps, he makes a great concession in admitting the existence of one supreme god, without acknowledging a multitude of deities subordinate to the first. For what the ancients can urge in defence of this obsolete opinion, I must beg leave to refer the reader to the dissertation prefixed to my translation of Orpheus; in addition to which let him attend to the following considerations. Is it possible that the machinery of the gods in Homer could be so beautiful, if such beings had no existence? Or can any thing be beautiful which is destitute of all reality? Do not things universally please in proportion as they resemble reality? Perhaps it will be answered, that the reverse of this is true, and that fiction more generally pleases than truth, as is evident from the great avidity with which romances are perused. To this I reply, that fiction itself ceases to be pleasing, when it supposes absolute impossibilities: for the existence of genii and fairies cannot be proved impossible; and these compose all the marvellous of romance. This observation is verified in Spencer’s Fairy Queen: for his allegories, in which the passions are personified, are tedious and unpleasant, because they are not disguised under the appearance of reality: while the magic of Circe, the bower of Calypso, the rocks of Scylla and Charybdis, and the melody of the Syrens, in the Odyssey of Homer, though nothing but allegories, universally enchant and delight, because they are covered with the semblance of truth. It is on this account that Mikon’s battles in heaven are barbarous and ridiculous in the extreme; for every one sees the impossibility of supposing gun-powder and cannons in the celestial regions: the machinery is forced and unnatural, contains no elegance of fancy, and is not replete with any mystical information. On the contrary, Homer’s machinery is natural and possible, is full of dignity and elegance, and is pregnant with the sublimest truths; it delights and enobles the mind of the reader, astonishes him with its magnificence and propriety, and animates him with the fury of poetic inspiration. And this, because it is possible and true.

[164]The sentence within the brackets is omitted in the Greek.

[165]That is, the circular form proceeds frombound, but right-lined figures frominfinity.

[166]That is, the number three.

[167]In Timæo.

[168]πρὸς ὃ, or,to which, is wanting in the original, and in all the published collections of the Zoroastrian oracles.

[169]That is Jupiter, who is called triadic, because he proceeds from Saturn and Rhea; and because his government is participated by Neptune and Pluto, for each of these is called Jupiter by Orpheus.

[170]This sentence, within the brackets, is omitted in the printed Greek.

[171]Fig. I. Fig. II.Thus let a part A E B cut off by the diameter A B (fig. I.) of the circle A E B D be placed on the other part A D B, as in fig. II. Then, if it is not equal to the other part, either A E B will fall within A D B, or A D B within A E B: but in either case, C E will be equal to C D, which is absurd.

[171]

Fig. I. Fig. II.

Thus let a part A E B cut off by the diameter A B (fig. I.) of the circle A E B D be placed on the other part A D B, as in fig. II. Then, if it is not equal to the other part, either A E B will fall within A D B, or A D B within A E B: but in either case, C E will be equal to C D, which is absurd.

[172]This objection is urged by Philoponus, in his book against Proclus on the eternity of the world; but not, in my opinion, with any success. See also Simplicius, in his third digression against Philoponus, in his commentary on the 8th book of Aristotle’s Physics.

[173]This definition is no where extant but in the commentaries of Proclus. Instead of it, in almost all the printed editions of Euclid, the following is substituted.A segment of a circle is the figure contained by a diameter, and the part of the circumference cut off by the diameter.This Mr. Simson has marked with commas, as a symbol of its being interpolated: but he has taken no notice of the different reading in the commentaries of Proclus. And what is still more remarkable, this variation is not noticed by any editor of Euclid’s Elements, either ancient or modern.

[174]As in every hyperbola.

[175]The Platonic reader must doubtless be pleased to find that Euclid was deeply skilled in the philosophy of Plato, as Proclus every where evinces. Indeed, the great accuracy, and elegant distribution of these Elements, sufficiently prove the truth of this assertion. And it is no inconsiderable testimony in favour of the Platonic philosophy, that its assistance enabled Euclid to produce such an admirable work.

[176]Concerning these crowns, or annular spaces, consult the great work of that very subtle and elegant mathematician Tacquet, entitledCylindrica et Annularia.

[177]In the preceding tenth commentary.

[178]This in consequence of every triangle possessing angles alone equal to two right.

[179]This too, follows from the same cause as above.

[180]Thus the following figure A B D C has four sides, and but three angles.

[181]The Greek in this place is very erroneous, which I have restored from the version of Barocius.

[182]For the Greek word ῥόμβος is derived from the verb ῥέμβω, which signifies to have a circumvolute motion.

[183]See the Orphic Hymns of Onomacritus to these deities; my translation of which I must recommend to the English reader, because there is no other.

[184]These twelve divinities, of which Jupiter is the head, are,Jupiter,Neptune,Vulcan,Vesta,Minerva,Mars,Ceres,Juno,Diana,Mercury,Venus, andApollo. The first triad of these is demiurgic, the second comprehends guardian deities, the third is vivific, or zoogonic, and the fourth contains elevating gods. But, for a particular theological account of these divinities, study Proclus on Plato’s Theology, and you will find their nature unfolded, in page 403, of that admirable work.

[185]For it is easy to conceive a cylindric spiral described about a right-line, so as to preserve an equal distance from it in every part; and in this case the spiral and right-line will never coincide though infinitely produced.

[186]As the conchoid is a curve but little known, I have subjoined the following account of its generation and principal property. In any given right line A P, call P the pole, A the vertex, and any intermediate point C the centre of the conchoid: likewise, conceive an infinite right line C H, which is called a rule, perpendicular to A P. Then, if the right line Apcontinued atpas much as is necessary, is conceived to be so turned about the abiding polep, that the point C may perpetually remain in the right line C H, the point A will describe the curve Ao, which the ancients called a conchoid.In this curve it is manifest (on account of the right line P O, cutting the rule in H that the pointowill never arrive at rule C H; but becausehO is perpetually equal to C A, and the angle of section is continually more acute, the distance of the point O from C H will at length be less than any given distance, and consequently the right line C H will be an asymptote to the curve A O.When the pole is at P, so that P C is equal to C A, the conchoid A O described by the revolution of P A, is called a primary conchoid, and those described from the polesp, andπ, or the curves Ao, Aω, secondary conchoids; and these are either contracted or protracted, as the eccentricity P C, is greater or less than the generative radius C A, which is called the altitude of the curve.Now, from the nature of the conchoid, it may be easily inferred, that not only the exterior conchoid Aωwill never coincide with the right line C H, but this is likewise true of the conchoids A O, Ao; and by infinitely extending the right-line Aπ, an infinite number of conchoids may be described between the exterior conchoid Aω, and the line C H, no one of which shall ever coincide with the asymptote C H. And this paradoxical property of the conchoid which has not been observed by any mathematician, is a legitimate consequence of the infinite divisibility of quantity. Not, indeed, that quantity admits of an actual division in infinitum, for this is absurd and impossible; but it is endued with an unwearied capacity of division, and a power of being diffused into multitude, which can never be exhausted. And this infinite capacity which it possesses arises from its participation of the indefinite duad; the source of boundless diffusion, and innumerable multitude.But this singular property is not confined to the conchoid, but is found in the following curve. Conceive that the right line A C which is perpendicular to the indefinite line X Y, is equal to the quadrantal arch H D, described from the centre C, with the radius C D: then from the same centre C, with the several distances C E, C F, C G, describe the arches El, Fn, Gp, each of which must be conceived equal to the first arch H D, and so on infinitely. Now, if the points H,k,l,n,p, be joined, they will form a curve line, approaching continually nearer to the right-line A B (parallel to C Y) but never effecting a perfect coincidence. This will be evident from considering that each of the sines of the arches H D,lE,nF, &c. being less than its respective arch, must also be less than the right-line A C, and consequently can never coincide with the right-line A B.But if other arches Di, Em, Fo, &c. each of them equal to the right-line A C, and described from one centre, tangents to the former arches H D,lE,nF, &c. be supposed; it is evident that the points H,i,m,o, &c. being joined, will form a curve line, which shall pass beyond the former curve, and converge still nearer to the line A B, without a possibility of ever becoming coincident: for since the arches Di, Em, Fo, &c. have less curvature than the former arches, but are equal to them in length, it is evident that they will be subtended by longer lines, and yet can never touch the right-line A B. In like manner, if other tangent arches be drawn to the former, and so on infinitely, with the same conditions, an infinite number of curve-lines will be formed, each of them passing between Hpand A B, and continually diverging from the latter, without a possibility of ever coinciding with the former. This curve, which I invented some years since, I suspect to be a parabola; but I have not yet had opportunity to determine it with certainty.

[186]

As the conchoid is a curve but little known, I have subjoined the following account of its generation and principal property. In any given right line A P, call P the pole, A the vertex, and any intermediate point C the centre of the conchoid: likewise, conceive an infinite right line C H, which is called a rule, perpendicular to A P. Then, if the right line Apcontinued atpas much as is necessary, is conceived to be so turned about the abiding polep, that the point C may perpetually remain in the right line C H, the point A will describe the curve Ao, which the ancients called a conchoid.

In this curve it is manifest (on account of the right line P O, cutting the rule in H that the pointowill never arrive at rule C H; but becausehO is perpetually equal to C A, and the angle of section is continually more acute, the distance of the point O from C H will at length be less than any given distance, and consequently the right line C H will be an asymptote to the curve A O.

When the pole is at P, so that P C is equal to C A, the conchoid A O described by the revolution of P A, is called a primary conchoid, and those described from the polesp, andπ, or the curves Ao, Aω, secondary conchoids; and these are either contracted or protracted, as the eccentricity P C, is greater or less than the generative radius C A, which is called the altitude of the curve.

Now, from the nature of the conchoid, it may be easily inferred, that not only the exterior conchoid Aωwill never coincide with the right line C H, but this is likewise true of the conchoids A O, Ao; and by infinitely extending the right-line Aπ, an infinite number of conchoids may be described between the exterior conchoid Aω, and the line C H, no one of which shall ever coincide with the asymptote C H. And this paradoxical property of the conchoid which has not been observed by any mathematician, is a legitimate consequence of the infinite divisibility of quantity. Not, indeed, that quantity admits of an actual division in infinitum, for this is absurd and impossible; but it is endued with an unwearied capacity of division, and a power of being diffused into multitude, which can never be exhausted. And this infinite capacity which it possesses arises from its participation of the indefinite duad; the source of boundless diffusion, and innumerable multitude.

But this singular property is not confined to the conchoid, but is found in the following curve. Conceive that the right line A C which is perpendicular to the indefinite line X Y, is equal to the quadrantal arch H D, described from the centre C, with the radius C D: then from the same centre C, with the several distances C E, C F, C G, describe the arches El, Fn, Gp, each of which must be conceived equal to the first arch H D, and so on infinitely. Now, if the points H,k,l,n,p, be joined, they will form a curve line, approaching continually nearer to the right-line A B (parallel to C Y) but never effecting a perfect coincidence. This will be evident from considering that each of the sines of the arches H D,lE,nF, &c. being less than its respective arch, must also be less than the right-line A C, and consequently can never coincide with the right-line A B.

But if other arches Di, Em, Fo, &c. each of them equal to the right-line A C, and described from one centre, tangents to the former arches H D,lE,nF, &c. be supposed; it is evident that the points H,i,m,o, &c. being joined, will form a curve line, which shall pass beyond the former curve, and converge still nearer to the line A B, without a possibility of ever becoming coincident: for since the arches Di, Em, Fo, &c. have less curvature than the former arches, but are equal to them in length, it is evident that they will be subtended by longer lines, and yet can never touch the right-line A B. In like manner, if other tangent arches be drawn to the former, and so on infinitely, with the same conditions, an infinite number of curve-lines will be formed, each of them passing between Hpand A B, and continually diverging from the latter, without a possibility of ever coinciding with the former. This curve, which I invented some years since, I suspect to be a parabola; but I have not yet had opportunity to determine it with certainty.

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