thoughts to the use which may be made of the properties of these rectangles and squares which he has been confidering: and as almost the simplest use, which can be made of magnitudes, is adding them to one another or taking them from one another; it is very manifeft that these properties will be in a manner useless, until we have acquired a readiness in adding them together, and taking them from one another: and this indeed is one of the most intricate and extenfive principles in geometry. Much of this bufinefs may, and indeed ought to be learnt from this very book; for whoever should treasure up in his memory whatever is contained in this book, it would be but useless lumber unless considered under this particular point of view. And fo much for the neceffity of this practice; but a few examples will beft explain my meaning. Suppofe in the second propofition you were required to add the rectangle contained by AB and BC to the rectangle contained by AB and AC; these taken together make the square of AB: again if from the fquare of AB you take away the rectangle contained by AB and BC; the remainder is the rectangle contained by AB and AC; all this is obvious from the infpection of the figure; but as these confequences may be wanted when the figure is not at hand for inspection, the reader ought to be able to derive them readily from any fingle ftraight line cut into two fegments. But again in the third propofition if you add the rectangle contained by AC and CB to the fquare of CB it makes the rectangle contained by AB and BC; or if you add the rectangle contained by AC and CB to the fquare of AC it makes the rectangle contained by AB and AC: And farther if from the rectangle contained by AB and BC you take away the fquare of BC there remains the rectangle contained by AC and CB. In the fourth propofition if you take the squares of AC and CB from the square of AB there is left the rectangle contained by AC and CB taken twice: or if the fquare of AC be added to the rectangle contained by AC and CB taken twice; their fum will be equal to the difference between the fquares of AB and BC; and this is also obvious from the infpection of the figure. The fifth propofition, in which is compared the rectangles made by the equal fegments of a line with those made by the unequal fegments, fegments, furnishes frequent opportunities for this kind of practice; thus if from the fquare of half the line BC you take away the square of CD the remainder is the the rectangle contained by AD and DB; and this square of CD added to the rectangle contained by AD and DB makes up the fquare of half the line. And in the fixth propofition if from the fquare of CD you take the fquare of CB the remainder is the rectangle contained by AD and DB. One might just observe upon the fifth; that the rectangle contained by AD and DB and the square of BC are bounded by the same extent of line; but that the spaces inclofed differ by the square of CD. It would be tedious to be more particular upon this fubject; I fhall therefore conclude with obferving that squares are generally added together or substracted from one another by the forty seventh propofition of the first book; and that they are doubled by the affistance of a right angled isofceles triangle as in the ninth and tenth propofitions; and may be halved upon the fame principles. CHA P. IV. In which is fhewn the abfurdity of applying numbers to illuftrate the propofitions in this book. IF you draw two indefinite straight lines at right angles to each other, and cut off from one, feven equal parts, beginning at the angular point, and from the other four of the fame equal parts, and compleat the right angled parallelogram; then, through each of the divifions, drawing lines parallel to the fides of the rectangle, the whole furface is divided into fquares all equal and twenty eight in number, which is the product of four multiplied by seven. And because this parallelogram is equal to any other upon the fame or an equal base and between the fame parallel lines ; and a triangle in the fame circumftances, the half of it: therefore this other parallelogram, though not divisible into fquares, is said to contain twenty eight fuch fquares; and the triangle fourteen of the fame. And as any rectilineal figure is divifible into triangles, upon thefe principles principles any rectilineal figure may have its contents expreft in square measure, inches feet or yards, according to the measure by which you fuppofe your lines to be divided. Now this way of expreffing the furface or contents of a right angled parallelogram in fquare measure by the product of the two fides about one of the right angles, has introduced a very abfurd practice upon this fecond book, which fome factious gentlemen have been pleafed to ftile illuftrations. But we shall be best able to decide how far this method of operation can be called an illuftration by confidering the ufe for which it is intended. Now the common affairs of life require, that we fhould pay a particular attention to small and large portions of extenfion within certain limits; which made it neceffary to affume particular parts of extenfion as a common meafure; and for this purpose at first certain parts of the human body feemed to have been used as is obvious from the names of meafures; an inch, a foot, afpan &c. and very probably he who obferved that the feet of different men were unequal in length, and thus proved the neceffity of having fome fixt standard measure, ftood pretty high in the opinion of his cotemporaries for ingenuity. Suppofe them to have made the standard measure which we call a foot; for fome purposes this would be too large and for others too fmall. The diftance of one place from another expreft in feet could bring no diftinct idea to the mind; it would be an improvement to take three feet and make an unit of that, under the different name of a yard, but still more to take one thousand feven hundred and fixty yards and call it a mile. And by fuch contrivances our perceptions of distance might be made tolerably accurate and suited to our circumstances. Surfaces are measured upon the fame plan, and expreft in square feet, yards or miles upon the principles explained above: and to the great comfort of the unthinking part of mankind all furfaces may be compared in a manner fufficiently accurate for the purposes of common life, not only without our ever attending to a furface; but what is more without our having any notion of extenfion. For our attention is turned to no particular kind of magnitude, as we are only to confider whether one number be greater than another; and here it is not neceffary to form your judgement by by the nature of numbers, but by a particular method of notation which you have been taught. It would be ftrange enough to put one upon measuring such figures as Euclid treats of in this fecond book, in order to get at those properties which he says belong to them; but it would be the greatest abfurdity to call this an illustration of Euclid's demonftrations; as the two methods are founded upon principles, in a certain sense directly contrary to each other: these mensurations drawing the reader's attention from Euclid's plan to fix it upon fomething else; or, more properly speaking, upon nothing. Neither is the conclufion scientifically accurate; for the lines taken at a venture cannot be divided according to any measure; particularly if the line be cut as is required in the eleventh propofition, it is impoffible to measure it and its parts. But to rest this matter entirely upon the most material objection, viz. that it is directly contrary to Euclid's ideas. He teaches how to turn any rectilineal figure into a fquare in the last propofition of this book. Which fhews that his plan is to make every ftep we take the object of the understanding, and therefore does not present a multitude of things to the mind which it is impoffible for it to comprehend, but two distinct things for its contemplation; he does not express irregular figures by a multitude of small squares of any particular name; but reduces any two, which he may have occasion to compare, to two squares; nor has his reasoning a reference to any particular. measure. In fhort if my intention was to purchase figures, I might trust to this practice; but if I meant to reafon about their properties it must be entirely laid aside. DISSERTATION V. HE triangle and rectangle two of the great inftruments of geometrical investigation, being confidered, Euclid proceeds to the third, which is of no less importance than either of the other two. And is here in its proper place, because the properties of the circle are derived, partly from the triangle, and partly from the rectangle. In the fecond book the propofitions are arranged according to their fimplicity; and not as in the first book according to the dependence which the propofitions have upon each other because the tenth might be read first, for any connexion which it has with the first nine propofitions; the four last indeed are connected with some of the preceeding propofitions, but none of the others. In this third book the arrangement is made partly according to the fimplicity of the properties and partly according to the connexion which they have with one another; and the explanation of this arrangement with fome remarks upon the method of demonftration shall be the subject of this dissertation. СНАР. І. Containing remarks upon the arrangement of the propofitions. IN the two firft books our author confiders the circle only as a mechanical instrument, and the use made of it, rests entirely upon the third poftulate; and unless it were introduced upon a different footing, it could hardly be reckoned a geometrical figure. In fhort in the first two books it is only a pair of compaffes with an indefinite extent and perfectly accurate. But here in this book |