ing uniformly in immaterial forms and impartible reasons, terminates and restrains the various motion of a fuperficies in its progreffion, and proximately unites its infinity. But in the images of thefe, when that which bounds fupervenes that which is bounded, it caufes, by this means, its limitation and bound. But if it should be enquired how lines are the extremities of every fuperficies, fince they are not the extremes of every finite figure; for the fuperficies of a sphere is terminated indeed, yet not by lines, but by itself? In answer to this, we must say, that by receiving a fuperficies fo far as it is diftant by a two-fold interval, we fhall find it terminated by lines according to length and breadth. But if we behold a fpherical fuperficies, we must receive it as that which is endued with figure; which poffeffes another quality, and conjoins the end with the beginning; and lofes its two extremities in the comprehenfive embraces of one: and this one extremity fubfifts in capacity only, and not in energy.

A Plane SUPERFICIES is that which is equally fituated between its bounding Lines.

IT

T was not agreeable to the ancient philofophers to establish a planè fpecies of fuperficies; but they confidered fuperficies in general, as the representative of magnitude, which is diftant by a two-fold interval. For thus the divine Plato* fays, that geometry is contemplative of planes, opposing it in division to stereometry, as if a plane and a fuperficies were the fame. And this was likewise the opinion of the demoniacal Ariftotle †. But Euclid and his followers confider fuperficies as a genus, but a plane as its fpecies, in the fame manner as rectitude of a line. And on this account he defines a plane feparate from a fuperficies, after the fimilitude of a right line. For he defines this last as equal to the space, placed between its points. And in like manner, he says, that two right lines being given, a plane fuperficies. occupies a place equal to the space fituated between thofe two lines.

VOL. I.

* Inv ii. De Rep.

In multis locis.

T

For

For this is equally fituated between its lines; and others alfo explaining the fame boundary, affert that it is conftituted in its extremities. But others define it as that to all the parts of which a right line may be adapted *. But perhaps others will fay, that it is the shortest of fuperficies, having the fame boundaries; and that its middle parts darken its extremities; and that all the definitions of a right line may be transferred into a plane fuperficies, by only changing the genus: fince a right, circular, and mixt line, commencing from lines, arrive even at folids, as we have afferted above; for they are proportionally, both in fuperficies and folids. Hence alfo, Parmenides fays, that every figure is either right, or circular, or mixt. But if you wish to confider the right in fuperficies, take a plane, to which a right line agrees in various ways; but if a circular receive a fpherical fuperficies; and if a mixt, a conic or cylindric, or fome one of that genus. But it is requifite (says Geminus) fince a line, and also a fuperficies is called mixt, to know the measure of mixture, because it is various. For mixture in lines, is neither by compofition, nor by temperament only: fince, indeed, a helix is mixed, yet one part of it is not straight, and another part circular, like those things which are mixed by compofition nor if a helix is cut after any manner, does it exhibit an image of things fimple, fuch as thofe which are mixed through temperament; but in these the extremes are, at the fame time, corrupted and confused. Hence, Theodorus the mathematician, does not rightly perceive, in thinking that this mixture is in lines. But mixture in fuperficies, is neither by compofition, nor by confusion; but subsists rather by a certain temperament. For conceiving a circle in a subject plane, and a point on high, and producing a right line from the point to the circumference of the circle, the revolution of this line will produce a conical fuperficies which is mixt. And we again refolve it into its fimple elements, by a parallel fection: for by drawing

This definition is the fame with that which Mr. Simfon has adopted inftead of Euclid's, expreffed in different words: for he fays, "a plane fuperficies is that in which any two points. being taken, the ftraight line between them lies wholly in that fuperficies." But he does not mention to whom he was indebted for the definition; and this, doubtless, because he confidered it was not worth while to relate the trifles of Proclus at full length: for these are his own words, in his note to propofition 7, book i. Nor has he informed us in what refpect Euclid's definition is indifline.

a fec

a fection between the vertex and the bafe, which fhall cut the plane of the generative right line, we effect a circular line. But the idea of lines, fhews that the mode of mixture is not by temperament; for neither does it fend us back to the fimple nature of elements: on the contrary, when fuperficies are cut, they immediately exhibit to us their producing lines The mode of mixture, therefore, is not the fame in lines and fuperficies. But as among lines there were fome fimple, that is, the right and circular, of which the vulgar alfo poffefs an anticipated knowledge without any previous inftruction; but the species of mixt lines require a more artificial apprehenfion: fo among fuperficies, we poffefs an innate notion of those which are especially elementary, the plane and spherical; but fcience and its reafon investigates the variety of those which are compofed through mixture. But this is an admirable property of fuperficies, that their mixture in generation is oftentimes produced from a circular line; and this also happens to a fpiral fuperficies. For this is understood by the revolution of a circle remaining erect, and turning itself about the fame point which is not its centre. And on this account, a fpiral alfo is threefold; for its centre is either in a circumference, or within, or external to a circumference. If the centre is in the circumference, a continued fpiral is produced: if within the circumference, an intangled one; if without, a divided one. And there are three fpiral fections correfponding to these three differences. But every spiral line is mixt, although the motion from which it is produced is one and circular. And mixt fuperficies are produced as well from fimple lines, (as we have faid,) while they are moved with a motion of this kind, as from mixt lines. Since, therefore, there are three conic lines, they produce four mixt fuperficies, which they call conoids. For a rectangular conoid, is produced from the revolution of the parabola about its axis: but that which is formed by the ellipfis, is called a fpheroid; and if the revolution is made about the greater axis, it is an oblong; but if about the leffer a broad fpheroid. Laftly, an obtufe-angled conoid is generated from the revolution of the hyper bola But it is requifite to know, that fometimes we arrive at the knowledge of fuperficies from lines, and fometimes the contrary; for from conical and fpiral fuperficies, we apprehend conical and spiral lines. Befides,

T 2

1

fides, this alfo must be previously received concerning the difference of lines and superficies, that there are three lines of fimilar parts (as we have already observed), but only two fuperficies, the plane and the fpherical. For this is not true of the cylindric, fince all parts of the cylindric fuperficies cannot agree to all. And thus much concerning the differences of fuperficies, one of which the geometrician having chosen (I mean the plane), this alfo he has defined; and in this, as a fubject, he contemplates figures, and their attendant paffions: for his discourse is more copious in this than in other fuperficies: fince, indeed, we may underftand right lines, and circles, and helixes in a. plane; also the sections of circles and right lines, contacts, and applications, and the conftructions of angles of every kind. But in other fuperficies, all these cannot be beheld. For how in one that is fpherical, can we apprehend a right line, or a right-lined angle? How, laftly, in a conic or cylindric fuperficies, can we behold fections of circles or right lines? Not undefervedly, therefore, does he both define this fuperficies, and difcufs his geometrical concerns, by exhibiting every thing in this as in a fubject; for from hence he calls the present treatise plane. And, after this manner, it is requifite to underftand that which is plane, as projected and conftituted before the eyes:. but cogitation as describing all things in this, the phantasy correfponding to a plane mirror, and the reasons refident in cogitation as dropping their images into its fhadowy receptacle.

*

+ A PLANE ANGLE, is the inclination of two Lines to each other in a Plane, which meet together, but are not in the fame direction..

SOM

OME of the ancient philofophers, placing an angle in the pre-dicament of relation, have faid, that it is the mutual inclination of lines or planes to each other. But others, including this in quality,

as

In the Greek words, but it fhould doubtless be read sxovas, images, as in the tranflation of Barocius.

Mr. Simfon, in his note on this definition, fuppofes it to be the addition of fome lefs skilful editor; on which account, and because it is quite ufclefs (in his opinion) he distinguishes

3

it

as well as rectitude and obliquity, fay, that it is a certain paffion of a fuperficies or a folid. And others, referring it to quantity, confess that it is a fuperficies or a folid. For the angle which fubfifts in fuperficies is divided by a line; but that which is in folids, by a fuperficies. But (fay they) that which is divided by thefe, is no other than magnitude, and this is not linear, fince a line is divided by a point; and therefore it follows that it must be either a fuperficies or a folid. But if it is magnitude, and all finite magnitudes of the fame. kind have a mutual proportion; all angles of the fame kind, i. e. which fubfift in fuperficies, will have a mutual proportion. And hence, the cornicular will be proportionable to a right-lined angle. But things which have a mutual proportion, may, by multiplication, exceed each other; and therefore it may be poffible for the cornicular to exceed a right-lined angle, which, it is well known, is impoffible,fince it is fhewn to be less than every right-lined angle. But if it is quality alone, like heat and cold, how is it divifible into equal parts? For equality, inequality, and divifibility, are not lefs refident in angles than in magnitudes; but they are, in like manner, essential. But if the things in which thefe are effentially inherent, are quantities, and not qualities, it is manifeft that angles alfo are not qualities. Since the more and the lefs are the proper paffions of quality *, but not equal and unequal. On this hypothefis, therefore, angles ought not to be called unequal, and this greater, but the other lefs; but they ought to be denominated diffimilars, and one more an angle, but the other lefs. But that these appellations are foreign from the effence of mathematical concerns, is obvious to every one: for every angle receives the fame definition, nor is this more an angle, but that lefs. Thirdly, if an angle is inclination, and belongs to the category of.

it from the rest by inverted double commas. But it is furely ftrange that the definition of angle. in general fhould be accounted ufelefs, and the work of an unfkilful geometrician. Such an affertion may, indeed, be very fuitable to a profeffor of experimental philofophy, who confiders the useful as infeparable from practice; but is by no means becoming a restorer of the liberal geometry of the ancients. Befides, Mr. Simfon feems continually to forget that Euclid was of the Platonic fect; and confequently was a philofopher as well as a mathematician. I only add, that the commentary on the prefent definition is, in my opinion, remarkably fubtle and accurate, and well deferves the profound attention of the greatest geometricians.

For a philofophical difcuffion of the nature of quality and quantity, confult the Commen taries of Ammonius, and Simplicius on, Ariflotle's Categories, Plotinus on the genera of beings,and Mr. Harris's Philofophical Arrangements.

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