or cross one another; but until they have met they are never confidered as containing an angle.

This plane rectilineal angle may be of three different kinds; according to the different inclinations which two ftraight lines may have to one another. Take two pens, and run a pin through the extremity of the one, and at the fame time, through the other, at fome distance from its extremity, fo that the one of the pens may make two angles with the other, on each fide one. Turn the pens into the fame direction, and making one of them revolve, and then obferve its progrefs in revolving until it takes the same direction on the other fide: and during this whole revolution there is but one fixed or determinate pofition of the lines with respect to one another; and that is when the revolving line has the fame inclination to the other on each fide; and thefe equal inclinations are called right angles; and the other two have their names according as the angle is greater or lefs than this.

The reader has been very inattentive to my directions upon this head, if he does not perceive, that there is an unlimited variety both of acute and obtufe angles, but that there is but one right angle.

And hence by the bye it is obvious that Euclid formed his definitions from common notions, as he reckons this one; that all right angles are equal, probably as a hint to the reader, that he might discover from what fource the definitions were derived; because this common notion may be regularly demonftrated from the definition of a right angle, and therefore must have been placed where it is for fome indirect purpofe. Upon the whole then we may conclude, that Euclid's definitions are not the imaginary phantoms which the metaphyficians have reprefented them: but notions derived from material objects, and new modeled, not for the purpose of empty fpeculations, but, that they might be applied to the fame kind of objects which first suggested them, upon a more accurate and enlarged plan.

CHAP.

THIS fcience has now acquired fuch an extent and accuracy, by improving the common notions into definitions; that our former instruments are now become inadequate to our purposes and views. For our notions both of a straight line and circle are fo refined and enlarged; that no inftrument is fufficiently accurate or extensive for performing these two problems, the drawing a straight line and the description of a circle.

Here the cafe of the science appears defperate; and the author reduced to the neceffity of giving up his high pretenfions to accuracy and univerfality, by fuffering the fcience to fall back again into its original state, as far as refpects its inftruments at least. And I am certain the artifice used upon this occafion to preserve the dignity of the science is by no means generally understood; and this will be confirmed beyond contradiction by attending to the foolish and childish reasons generally given for admitting these poftulates, alledging that it is so easy to conceive how they may be done, that they are to be admitted without the leaft fcruple or hesitation, That a thing may be easily done is furely a very bad reafon for neglecting to give rules for doing it: and if that had been the case in the present instance, I am pretty confident thefe commentators would have been faved the trouble of their apology. But the opinion of Newton I fuppofe will be decifive upon this occafion, and we find him expreffing himself directly to our purpose as follows. "Nam et linearum rectarum et circulorum defcriptiones, in quibus geometria fundatur, ad mechanicam pertinent. Has lineas defcribere geometria non docet fed poftulat. Poftulat enim ut tyro eafdem accurate defcribere prius didiceret, quam limen attingat geometria; dein, quomodo per has operationes problemata folvantur, docet; rectas et circulos defcribere problemata funt, fed non geometrica. Ex mechanica poftulatur horum folutio, in geometria docetur folutorum ufus. Ac gloriatur geometria quod, tam paucis principiis aliunde petitis, tam multa præstet.

After fuch an authority the judicious reader will be ready to agree with me, that these problems are taken for granted, not

because

cause it is easy to conceive how they may be done, but for a much better reason, because it is impoffible to do them by any method consistent with the principles of this science.

But even in this, which must have been a matter of no small difficulty, our author has done every thing in the power of man, to render this unpromiffing part as correct as poffible, by making it put on a very respectable figure indeed, giving it such an air, as discovers but little of its mechanic original,

An ordinary genius would probably have given directions for drawing a straight line and defcribing a circle, with a ruler and compaffes; and when the limited nature of his inftruments was mentioned as an objection to his fcience, and that he could not with propriety extend his conclufions beyond a sheet of paper; he would then probably propose a chain and poles by the affistance of which he might undertake to enlarge the scale of his operations; yet with waggon loads of fuch inftruments he never could hope to produce one fcientific conftruction.

But Euclid has acted with more judgement, and never would be acceffory to the opening a door for admitting upon his fcientific ftage the whole tribe of mechanics, with the several implements of their trade. He throws them two problems to perform, but this is to be done behind the scenes, without exhibiting their mechanical inftruments, declaring that upon every future is determined to ftand or fall by his own principles.

emergency, he

Of the inftruments made ufe of for communicating geometrical knowledge.

ALTHOUGH language be a general medium for communicating knowledge of every kind; yet particular fubjects require the introduction of some auxiliary inftruments. Unless the nature of the thing is fuch, that it cannot fail of itself to make a strong impreffion upon the mind, or that it is of no great consequence whether it be particularly examined or not, it will be found difficult to command the attention; and particularly, which is absolutely neceffary in a chain of reasoning, to make the thoughts of VOL. I.

C

the

the reader and writer follow each other in the fame order, with a certainty that they are both thinking of the fame thing at the fame time. The art of logic is an inftrument of this kind, but the great. diverfity of opinions prevailing among mankind is a fufficient proof that this inftrument is either neglected or misapplied. Indeed that wonderful uniformity of opinion which has been univerfally found among all nations, and in all the different ages of the world upon geometrical fubjects, cannot fail to draw our attention to this fingular circumftance; and make the learner defirous to be let into the fecret of this harmony, as far as one ignorant of the subject can be supposed capable of entering. And this will be found to proceed from the method of demonftration to which geometricans have tied themselves down; and their being poffeffed of an inftrument, perfectly adequate to the purpose of communicating their knowledge to others, and by which they are able infallibly to examine their And this inftrument in geometry is figures or diagrams: which Euclid exprefes by the different names of χήμα, καταγραφή, d. The first is his general term, the fecond he commonly applies to a figure which has been described or at least compleated; and the third commonly to fimilar figures.

own.

Now these diagrams or figures may be made fo exactly to refemble the subject matter of any propofition, that, if we think at all, it is impoffible to mistake the order of thinking, which the author has prescribed, or to draw a different conclufion from that, to which he intends to lead us. I fay a different; because the figure itself may lead us to a limited or partial conclufion; the remedy for which I shall explain at length in the next differtation.

The fimpleft kind of rectilineal figure is the triangle; the dif ferent parts of which the learner ought to be familiarly acquainted with; and to keep in his mind, that befides the triangular space itself; there are fix different magnitudes which go to the making it up; viz. three fides and three angles; each of which ought to be particularly attended to. It will be useful alo to obferve; that the triangles ABG, ACF though they have feveral parts in common, are nevertheless to be confidered as much as diftinct and different triangles, as if feparated from each other by the distance of a thousand miles.

The.

The three fides of the triangle ABG are AG, AB, BG and the three angles are ABG, AGB, GAB: and of the triangle ACF the three fides are AF, AC, CF; and the three angles ACF, AFC, CAF and the angle at A or DAE is faid to be common to the two triangles; and therefore they have one angle equal to one angle; and it is faid to be contained by the fides FA, AC when it is confidered as an angle of the triangle ACF: but when it is confidered as an angle of the triangle ABG it is said to be contained by GA, AB.

and the

Again FBC and BCG are alfo two distinct triangles three fides of the triangle FBC are BF, FC, CB and the three angles BFC, FCB, CBF: and the three fides of the triangle BCG are CG, GB, BC; and the three angles CGB, GBC, BCG: these triangles have a common fide viz. the fide BC; and this fide is said to be extended under the angles BFC and BGC. These things ought to be well understood and strongly impreffed upon the memory.

But farther, we find that the two triangles ACF and BFC have a fide and an angle common to both, viz. the fide FC and the angle AFC which is the angle BFC; also the two triangles ABG, BCG have a fide and an angle common to both, viz. the fide BG and the angle AGB which is the angle CGB.

I have now explained at some length the original and nature of the geometrical principles; and the inftruments made use of for communicating this kind of knowledge, taking notice at the fame time of some of the most unusual forms of expreffion and which are apt to perplex the learner at his first setting out.