name, defcribing trapeziums them felves, according to the difference of thofe four figures, in which the property of parallelograms is verified. And this is to have the oppofite fides and angles equal. For a quadrangle and an oblong, and a rhombus, have their oppofite. fides and angles equal. But in a rhomboides he only adds this, that its oppofite fides are equal, left he should define it by negations alone, fince he neither calls it equilateral, nor rectangular. For where we want proper appellations, it is neceffary to ufe fuch as are common. But we should hear Euclid fhewing that this is common to all paralletograms. But a rhombus appears to be a quadrangle having its fidesmoved, nnd a rhomboides a moved oblong. Hence, according to fides, these do not differ from those; but they vary only according to the obtufeness and acutenefs of angles; fince the quadrangle and the oblong are rectangular. For if you conceive a quadrangle or an oblong, having its fides drawn in fuch a manner, that while two of its oppofite angles are dilated, the other two are contracted; then the dilated angles will appear obtufe, and the contracted, acute. And the appellation of rhombus † feems to have been impofed from motion. For if you conceive a quadrangle moving after the manner of a rhombus, it will appear to you changed in order, according to its angles :just as if a circle is moved after the manner of a fling, it will immediately exhibit the appearance of an ellipfis. But here you may perhaps enquire concerning the quadrangle, why it has this denomination? and why the appellation of quadrangle may not be applied to other quadrilateral figures, as the name of triangle is common to all those which are neither equiangular nor equilateral, and in like manner of quinquangles or pentagons; for the geometrician, in these, adds only the particle an equilateral triangle, or a quinquangle, which is equilateral and equiangular, as if thefe could not be otherwife than fuch as they are? But when he mentions a quadrangle, he immediately indicates that it must be equilateral and rectangular. But the reason of this is as follows: a quadrangle alone has the beft fpace, both

*The Greek in this place is very erroneous, which I have reftored from the verfion of Barccius.

For the Greek words is derived from the verb gew, which fignifies to have a circumvolute metion.

VOL. I.

A.a

according

according to its fides and angles. For each of the latter is right, intercepting a measure of angles, which neither receives intention nor remiffion. As it excels, therefore, in both refpects, it defervedly obtains a common appellation. But a triangle, though it may have equal fides, yet will in this cafe have all its angles acute, and a quinquangle all its angles obtufe. Since, therefore, of all quadrilateral figures, a quadrangle alone is replete with equality of fides, and rectitude of angles, it was not undeservedly allotted this appellation: for, to excellent forms, we often dedicate the name of the whole. But it appeared alfo to the Pythagoreans, that this property of quadrilateral figures, principally conveyed an image of a divine effence. For they particularly fignified by this, a pure and immaculate order. Since reЯitude imitates inflexibility, but equality a firm and permanent power for motion emanates from inequality, but quiet from equality itself. The gods, therefore, who are the authors to all things of stable difpofition, of pure and uncontaminated order, and of indeclinable power, are deservedly manifefted as from an image, by a quadrangular figure. But, befides thefe, Philolaus alfo, according to another apprehenfion, calls a quadrangular angle, the angle of Rhea, Ceres and Vesta. For, fince a quadrangle conftitutes the earth, and is its proximate element, as we learn from Timæus, but the earth herself receives from all these divinities, genital feeds, and prolific powers, he does not unjustly confecrate the angle of a quadrangle to these goddesses, the bestowers of life. For fome call both the earth and Ceres, Vesta *, and they say that Rhea totally participates her nature, and that all generative causes are contained in her effence. l'hilolaus, therefore, fays that a quadrangular angle comprehends, by a certain terreftrial power, one union of the divine genera. But fome affimilate a quadrangle to univerfal virtue, fo far as every quadrangle from its perfection has four right angles. Juft as we fay that each of the virtues is perfect, content with itself, the measure and bound of life, and the middle of every thing which, in morals, correfponds to the obtufe and acute. But it is by no means proper to conceal, that Philolaus attributes a triangular angle to four, but a quadrangular angle to three gods, ex

See the Orphic Hymns of Onomacritus to thefe deities; my tranflation of which I muft recommend to the English reader, because there is no other.

hibiting

1

hibiting their alternate tranfition, and the community of all things in all, of odd natures in the even, and of even in the odd. Hence, the tetradic ternary, and the triadic quaternary, participating of prolific and efficacious goods, contain the whole ornament of generable natures, and preferve them in their proper ftate. From which the duodenary,. or the number twelve, is excited to a fingular unity, viz. the government of Jupiter. For Philolaus fays, that the angle of a dodecagon(or twelve-fided figure) belongs to Jove, fo far as Jupiter contains and preferves, by his fingular union, the whole number of the duodenary. For alfo, according to Plato, Jupiter prefides over the duodenary and governs and moderates the univerfe with abfolute fway. And thus much we have thought proper to difcourfe concerning quadrilateral figures, as well declaring the fenfe of our author, as likewise affording an occafion of more profound infpections to fuch as defire. the knowledge of intelligible and occult effences.

*

PARALLEL RIGHT LINES are fuch as being in the fame Plane, and produced both ways infinitely, will in no part mutually coincide.

WHA

HAT the elements of parallels are, and by what accidents inthefe they may be known, we shall afterwards learn: but what parallel right lines are, he defines in these words: "It is requifite, therefore (fays he), that they fhould be in one plane, and while they are produced both ways have no co-incidence, but be extended in infinitum." For non-parallel lines alfo, if they are produced to a certain. diftance, will not coincide. But to be produced infinitely, without coincidence, expreffes the property of parallels. Nor yet this abfolutely, but to be extended both ways infinitely, and not coincide.

* Thefe twelve divinities, of which Jupiter is the head, are, Jupiter, Neptune, Vulcan, Feta, Minerva, Mars, Cères, Juno, Diana, Mercury, Venus, and Apollo. The first triad of thefe is demiurgic, the fecond comprehends guardian deities, the third is vivific, or zoogoric, and the fourth contains elevating gods. But, for a particular theological account of thefe divinities, ftudy Proclus on Plato's Theology, and you will find their nature unfolded, in page 403, of that admirable work.

For it is poffible that non-parallel lines may alfo be produced one way infinitely, but not the other; fince, verging in this part, they are far diftant from mutual coincidence in the other. But the reafon of this is, because two right-lines cannot comprehend fpace; for if they verge to each other both ways, this cannot happen. Besides this, he very properly confiders the right-lines as fubfifting in the fame plane. For if the one should be in a subject plane, but the other in one elevated, they will not mutually coincide according to every position, yet they are not on this account parallel. The plane, therefore, fhould be one, and they fhould be produced both ways infinitely, and not coincide in either part. For with these conditions, the right-lines will be parallel. And agreeable to this, Euclid defines parallel right-lines. But Poflidonius fays, parallel lines are fuch as neither incline nor diverge in one plane; but have all the perpendiculars equal which are drawn from the points of the one to the other. But fuch lines as make their perpendiculars always greater and lefs, will fome time or other coincide, because they mutually verge to each other. For a perpendicular is capable of bounding the altitudes of spaces, and the distances of lines. On which account, when the perpendiculars are equal, the distances of the right lines are also equal; but when they are greater and lefs, the diftance alfo becomes greater and lefs, and they mutually verge in those parts, in which the leffer perpendiculars are found. But it is requifite to know, that non coincidence does not entirely form parallel lines. For the circumferences of concentric circles do not coincide: but it is likewife requifite that they should be infinitely produced. But this property is not only inherent in right, but alfo in other lines: for it is poffible to conceive fpirals defcribed in order about right lines, which if produced infinitely together with the right lines, will never coincide *. Geminus, therefore, makes a very proper divifion in this place, affirming from the beginning, that of lines fome are bounded, and contain figure, as the circle and ellipfis, likewife the ciffoid, and many others; but others are indeterminate, which may be produced infinitely, as the right-line, and the section of a right-angled, and

For it is eafy to conceive a cylindric fpiral deferibed about a right-line, fo as to preferve an equal distance from it in every part; and in this cafe the fpiral and right-line will never Coincide though infinitely produced.

3

obtufe

obtufe angled cone; likewife the conchoid itself. But again, of thofe which may be produced in infinitum, fome comprehend no figure, as the right-line and the conic fections; but others, returning into them-. felves, and forming figure, may afterwards be infinitely produced. And of these fome will not hereafter coincide, which refift coincidence, how far foever they may be produced; but others are coincident, which will fome time or other coincide. But of non-coincident lines, fome are mutually in one plane; and others not. And of non-coincidents fubfifting in one plane, fome are always mutually distant by an equal interval; but others always diminish the interval, as an hyperbola in its inclination to a right-line, and likewise the conchoid *. For thefe, though

* As the choncoid is a curve but little known, I have fubjoined 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 choncoid: likewife, conceive an infinite right line CH, which is called a rule, perpendicular to A P. Then, if the right line Ap continued at p as much as is neceflary, is conceived to be so turned about the abiding pole p, that the point C may perpetually remain in the right line CH, the point A will deferibe the curve Ao, which the ancients called a conchoid.

In this curve it is manifeft (on account of the right line PO, cutting the rule in H that the point will never arrive at rule CH; but becaufe O is perpetually equal to CA, and the angle of fection is continually more acute, the distance of the point O from CH will at length be lefs than any given distance, and confequently the right line CH will be an allymptote to the curve A O.

When