for there is no circumftance given by which we can determine which fide is the longest; there ought therefore to be added to the prefent fuppofition, AC being longer than AB; but the determination of this point I shall leave to the curious. But farther I would even recommend the fame kind of examination, where there was no probability of finding any variation; as for instance in the fixteenth propofition, the conclufion depends upon FCD's being an angle; for if that could ever cease to be an angle, the outward angle might be equal to the inward oppofite one; but that is impoffible because then the point F would be in BC produced; and confequently the point E upon C that is the middle of a line at its extremity, which is abfurd. Having been thus particular in fhewing what is not Euclid's method of demonstrating, I fhall conclude this chapter with a few inftances tending more directly to explain what it is. And his great peculiarity feems to be a determined refolution always to refer directly to fome principle, and never trust to a vague conception; or more properly never to make use of a vague expreffion: The thirteenth propofition is a remarkable instance to this purpose, That the angles are equal to two right angles ftrikes the senses immediately and produces a conviction which has something fluctuating in the nature of it; arifing from observing that the angle ABD is above a right angle, by just as much as ABC is less than one; now this is not a principle fufficiently diftinct to reason from in a demonstrative science; and Euclid has shewn great art in the demonftration of this very propofition; which he has reduced to the common notion, magnitudes which are equal to the fame are equal to one another; and how accurately he keeps up, through the whole demonstration, to the notion of the angles being magnitudes, cannot fail to engage the attention of a judicious reader. The twentieth propofition furnishes an instance to the fame purpose: every one believes that two fides of a triangle are greater than the third; and he may take up this opinion from a confideration of many different circumftances; he may confider that a ftraight line muft furely be the fhortest way between two points or he may truft to the judgement of the afs of the Epicureans; when a bundle of hay is placed at one of the angles; but VOL. I. i ftill still he will find fomething unfettled in his conviction as resting upon no determinate principles; and Euclid never leaves any thing which he takes in hand to demonftrate, in this unfettled ftate: he makes this property follow, not from any unstable principles or random conceptions, but from the very nature of a triangle; and a thorough examination, of the whole appartus neceffary for the demonstration of this principle, will give the intelligent reader a wonderful insight into Euclid's method of demonstration. It seems almost a defperate undertaking, to endeavour at reviving a tafte for accurate demonstration; because the common vague conceptions of magnitude, proped by a ruler and compaffes, when they begin to totter, are judged capable of performing every thing that can be expected from this fcience. But if a reformation is at all practicable, it seems most likely to be brought about, by turning the attention of the world to Euclid's method of demonftration ; and I ground my hopes of fuccefs upon two circumstances; namely that the modern writers of elements of Geometry neither underftand the nature of their own demonstrations, nor the force of thofe of Euclid. Because if they understood the force of Euclid's they would be ashamed to dignify their own with such a name; and if they understood the nature of their own, they must perceive that a great deal of expence both of time and thought might be faved by delivering their propofitions as facts, which had been demonstrated; unless indeed we fuppofe that what they call demonstrations are added merely in compliance to vulgar prejudice. Now could a judicious reader, no ways interested in the iffue of the debate, be once brought to weigh the merits of fuch demonftrations; he muft either decide in favour of fuch as Euclid's, or declare that all demonftration was now become unneceffary; and that therefore the farce of it fhould be laid afide. But But upon fuch an event, fuch is the charm of truth, I would naturally expect a confiderable number to leave their new mafters, as foon as they had relinquished their pretenfions to demonstration: and it is upon fuch hopes only that I proceed to give fome other inftances as characteristical of Euclid's method of demonftration. Euclid's demonftration of the feventh propofition, may be compared with the one given in Defchales's edition of Euclid; which was was no doubt intended as an improvement in some respect or other. And this feems to me the more extraordinary because the dømon-/e stration of the seventh is remarkably distinct and pertinent; leading to the conclusion so directly that the full force of the demonstration must be perceived by the mind and I am perfuaded that our author would have given up all thoughts of writing his elements, if he had found himself obliged to rest such a material part of his science upon fo unstable a foundation, as the description of two circles, with an indeterminate radius and even before it could be known in how many points their circumferences would cut each other; because every point of intersection, that did not lie on the other fide of AB would contradict the propofition; and prove that two triangles might be placed according to the fuppofition, until these points were reduced to one, which is not so easy to do as a fuperficial reader may imagine. But there is nothing which I have obferved Euclid more cautiously to shun than the describing a circle with an indeterminate radius; or more properly, than the fixing a line, otherwise undetermined, for the purpose of defcribing a circle. It is true he makes use of confequences which must be derived from circles defcribed in this manner; but this is fuppofed to be transacted in fuch a manner as to bring no difgrace upon the fcience. In the fixteenth propofition AC the fide of any triangle is required to be cut in halves, which cannot be done until the line is fixed; but then I immediately perceive that, notwithftanding this, as no confequence is deduced from AC's having any particular length, the demonstration is nevertheless general; fo that I can finish my construction and have AC as undetermined as I found it; fo that this tranfaction is fuppofed to be carried on in private by the reader himself; and which he acquieses in, as bringing no reflexion upon the science. There is another thing which I fhall just observe before finishing this chapter: Euclid's demonstrations are often more general than they feem to be; for whenever, it would be only a repetition of the fame steps, he always omits them, whether in a conftruction or demonftration. The forty fifth propofition furnishes a remarkable inftance of this; he fhews how to turn a fourfided figure into a parallelogram &c. and as no new circumstance i 2 would would occur, whatever be the number of fides of the figure; he concludes that it is general, without adding a word more; which has misled fome ignorant conceited people to think that they improve upon Euclid by fhewing a fimpler way of turning a fourfided figure into a parallelogram; never confidering that Euclid has turned any rectilineal figure into a parallelogram, whatever be the number of its fides; and whatever be their position to one another. Now I hope it is evident that Euclid had some other plan in his demonstrations than to convince the obftinate sophists; and what that plan is I fhall explain in the next chapter. CHA P. II. Of the arrangement of Euclid's propofitions. THOSE who know nothing of geometry tell us the science should be delivered according to the following arrangement. We fhould first begin with a point; and after having laid before the reader all its various properties, such as position, want of dimenfions and the like, we are next to proceed to the ftraight line; and after delivering every thing that can be faid of a single straight line, we may then advance a step farther and take two of them under our confideration; firft enquiring into the properties of such as are parallel; and this subject being exhausted we may next examine two straight lines that meet so as to form an angle; which will furnish us with a fruitful source of speculation; because this will now be the proper place to treat of angles; to shew how they may be proved to be equal, and into how many kinds they ought to be divided, not forgetting to prove that a right angle confifts of ninety degrees. Now fuppofing the properties of straight lines exhausted, the natural transition is to fuch as are crooked, arranging them into different orders, beginning with the circle; but take care not to confider it as a figure; for that would destroy all order; for it is the circumference only that belongs to our present subject: we may next proceed to the conic fections, and fo on to other lines. The The business of lines being finished, there would be abundant matter for dispute, when we came to treat of figures, whether we should begin with the triangle or the circle; with the triangle as being the fimpleft rectilineal figure; or with the circle as being bounded by one fingle line; the circle might probably have the preference upon a full examination as being a perfect figure, and for other reasons which it would be too tedious to infift upon at prefent. But fuppofing this difficulty got over; the triangle would certainly claim the next place; and of all the different kinds of triangles, the equilateral would put in for the first; the isofceles would deserve to be confidered next; from the ifofceles the tranfition must be to the scalene; taking care all the time, in delivering their properties to fay nothing of their angles, for that would be to confound the nature of things; because they are afterwards to be divided according to their angles, right, acute and obtuse; and this new divifion would furnish the proper place for that part of the subject. From triangles the most natural progress would be to the fquare, and then to figures lefs regular. And this is the true philofophical arrangement. Such are the engines which Folly has been erecting for battering this noble work of our author; feconded with other auxiliary schemes, the force of which feems leveled against it, though more indirectly. But what holds the greatest vogue at present, as falling in with our indolent disposition, and bids fair for driving him to the laft extremity, is the propofal of confidering no more of the subject than what is abfolutely neceffary; with the inviting title; as applied to fuch and fuch useful purposes of life. Elements of geometry carefully weeded of every propofition tending to demonftrate another; all lying fo handy, that you may pick and chuse without ceremony. This is useful in fortification: you cannot play at billiards without this. You only look through a telescope like a Hottentot. until this propofition is read; with many fuch powerful strokes of Rhetoric to the fame purpose. And upon fuch terms, and with fuch inducements who would not be a mathematician? Who would go to work with all that apparatus which I have described as neceffary for understanding Euclid; when he has only to take a pleafing walk with Clairaut upon the flowery banks of fome delightful! |