Previous to his de

although true, is entirely common.

monstration he fuppofes two fquares described, the one circum

of unbending a circular line. For fince the fame cause, acting every where fimilarly and equally, produces every where fimilar and equal effects; and the arch go, is every where VOL. I.

h

equally

circumfcribing the circle, which will be confèquently greater; the other infcribed, which will be. confequently less than the given circle. Hence, becaufe the circle is a medium between the two given fquares, let a mean fquare be found between them, which is eafily done from the principles of geometry; this mean fquare, Bryfo affirms, fhall be equal to the given circle. In order to prove this, he reasons after the following manner: thofe things which compared with others without any respect, are either at the fame time greater, or at the fame time lefs, are equal among themselves: the circle and the mean square are, at the fame time, greater than the internal, and at the fame time less than the external fquare; therefore they are

equally remitted or unbent, it will defcribe a line fimilar in every part. Now, on account of the fimplicity of the impulfive motion, fuch a line muft either be straight or circular; for there are only three lines every where fimilar, ie. the right and circular line, and the cylindric helix; but this laft, as Proclus well obferves in his following Commentary on the fourth definition, is not a fimple line, because it is generated by two fimple motions, the rectilineal and circular. But the line which bounds more than two equal tangent arches cannot be a right line, as is well known to all geometricians; it is therefore a circular line. It is likewife evident, that this arch ox is concave towards the point g: for if not, it would pass beyond the chord ox, which is abfurd. And again, no arch greater than the quadrant can be unbent by this motion for any one of the radii, as a p.beyond go, has a tendency from, and not to the tangent gx, which laft is neceffary to our hypothefis. Now if we conceive another quadrantal ach of the circle go e f, that is gy, touching the former in g to be unbent in the fame manner, the arch y fhall be a continuation of the arch a o; for if γ xx be drawn perpendicular to g, as in the figure, it shall be a tangent in to the equal arches yx, xo; because it cannot fall within either, without making the fine of fome one of the equal arches, equal to the right-line xg, which would be abfurd. And hence we may eafily infer, that the centre of the arch y xo, is in the tangent line ag. Hence too, we have an eafy method of finding a tangent right-line equal to a quadrantal arch: for having the points y, o given, it is eafy to find a third point, as s; and then the circle paffing through the three: points o, s, y, fhall cut off the tangent xg, equal to the quadrantal arch go. And the points may be fpeedily obtained, by defcribing the arch gs with a 1adius, having to the radius ag the proportion of 6 to 4; for then g s is the fixth part of its whole circle, and is equal to the arch g. And thus, from this hypothefis, which, I prefume, may be as. readily admitted as the increments and decrements of lines in fluxions, the quadrature of the circle may be geometrically obtained; for this is cafily found, when a right-line is difcovered equal to the periphery of a circle. I am well aware the algebraifts will confider it as useless,. because it cannot be accommodated to the farrago of an arithmetical calculation; but I hope the lovers of the ancient geomary will deem it deferving an accurate inveftigation; and if they can find no paralogifin in the reafoning, will confider it as a legitimate demonftration.

equal

equal among themselves. produce science, because it is built only on one common principle, which may with equal propriety be applied to numbers in arithmetic, and to times in natural science. It

This demonftration can never

is defective, therefore, because it affumes no principle peculiar to the nature of the circle alone, but fuch a one as is common to quantity in general.

13. It is likewife evident, that if the propofitions be univerfal, from which the demonstrative fyllogifm confists, the conclusion must neceffarily be eternal. For neceffary propofitions are eternal; but from things neceffary and eternal, neceffary and eternal truth muft arife. There is no demonstration, therefore, of corruptible natures, nor any science abfolutely, but only by accident; because it is. not founded on that which is univerfal. For what confirmation can there be of a conclufion, whose fubject is diffoluble,. and whose predicate is neither always, nor fimply, but only partially inherent? But as there can be no demonstration, fo likewise there can be no definition of corruptible natures; because definition is either the principle of demonstration, or demonstration differing in the position of terms, or it is a certain conclufion of demonftration. It is the beginning

of demonftration, when it is either affumed for an immediate propofition, or for a term in the propofition; as if any one should prove that man is rifible, becaufe he is a rational animal. And it alone differs in pofition from demonftration, as often as the definition is fuch as contains the cause of its fubjects existence. As the following: an eclipfe of the fun is a concealment of its light, through the interpofition of the moon between that luminary and the earth. For the order of this definition being a little changed, paffes into a demonftration: thus,

The moon is fubjected and oppofed to the fun :

That which is fubjected and oppofed, conceals :

The moon, therefore, being fubjected and oppofed, conceals the fun.

But that definition is the conclufion of demonftration,

DEMONSTRATIVE SYLLOGISM.

Ixi

which extends to the material caufe; as in the preceding inftance, the conclufion affirming that the fubjection and oppofition of the moon conceals the fun, is a definition of an eclipse including the material cause.

Again, we have already proved that all demonftration con-fifts of fuch principles as are prior in the nature of things;: and from hence we infer, that it is the business of no fcience to prove its own principles, fince they can no longer be called principles if they require confirmation from any thing prior to themselves; for, admitting this as neceffary, an infinite series of proofs must enfue. On the contrary, if this be not neceffary, but things most known and evident are admitted, these must be conftituted the principles of science. He who poffeffes a knowledge of these, and applies them as mediums of demonftration, is better fkilled in. fcience, than he who knows only pofterior or mediate propofitions, and demonstrates from pofterior principles. But here a doubt arifes whether the first principles of geometry, arithmetic, mufic, and of other arts, can ever be demonfrated? Or fhall we allow they are capable of proof, not by that particular fcience which applies them as principles or caufes of its conclufions? If fo, this will be the office of fome fuperior feience,-which can be no other than the first philosophy, to whose charge the task is committed; and whose univerfal embrace circumfcribes the whole circle of fcience, in the fame manner as arithmetic comprehends mufic, or geometry optics.This is no other than that celebrated wisdom which merits the appellation of science in a more fimple, as well as in a more eminent degree than others: not, indeed, that all caufes are within its reach, but fuch only as are the principal and the best, because no cause fuperior to them can ever be found. Hence the difficulty of knowing whether we poffefs fcience or

not,