2R-P must be employed as elements in its actual calculation. These two sets are not necessarily the same, either in number or kind. Take, for example, Professor Leslie's case, where c: (a, b, P). It is quite true that when a, b, P are fixed, c is fixed. But proceed to the actual calculation of e, and very different things appear upon the scene. For the value of c has to be collected from the well-known trigonometrical formula, that a+b: a-b :: tang of: : tang. of semi-difference of the angles at the 2 base c. Here then, in lieu of the solitary and heterogeneous angle P, start up among the practical elements of the calculation two straight lines, in the shape of the tangents of two arcs, which of course do not afterwards fail to conduct themselves with perfect submission to the law of homogeneity. And with all this the proposers of the analytical proof are bound to make their argument square; for the concession of their own demands ends in establishing the results of vulgar trigonometry, and not in altering them. On the whole, therefore, the pretence of knowing what quantities must be ejected to preserve the law of homo geneity, is visionary till it is known what quantities may or may not sul sequently appear among the practical elements of the calculation; which is impossible in the preliminary stage. The point, then, which the supporters of the analytical proof will be called on to establish, is why the possibility of the apparition of new elements which is visible in other cases (and which in Professor Leslie's case they actually claim by demanding the admission of R), is necessarily non-existent in their own. Take, for example, the hyperbolic triangle A HG, towards the end of 16 of this collection. In this it is plain that if the line A B and the straight lines AG and GH are fixed and determined, the angle A H G must be one certain angle and no other. But proceed to calculate the comparative magnitude of the angle to different values of a G, and there immediately start into action new elements in no stinted number, viz., two constant straight lines under the denominations of a major and a minor axis, and a varying straight line under the title of abscissa, to say nothing of the radius of a circle and such sines or tangents of different arcs thereof as may be found necessary in the process. How then do the opponents know that there are no more elements in the other case? If nature had contrived that the three angles of a triangle should not be always equal to two right angles, the proportionality of the sides of similar triangles would not have held good, and in making Tabies (for example) of the tangents to different arcs of a circle, the magnitude of the radius of the circle must in some guise or other have been an element. The tangent of 45° to a radius of one foot would have borne some given ratio to a foot, and the tangent of the same angle to a radius of two feet, instead of bearing the same ratio to two feet, would have borne some different one. There must have been a column of numbers to be applied according to the length of the radius, to obtain the true tangent of the angle to a given length of radius; in the same manner as would be necessary if it was desired to frame a Table for finding the perpendiculars in the hyperbolic triangle for different lengths of base. In other words, there would have been more elements. That this is not so, may be a happy event; but by what evidence included in their proposed demonstration, do they know that it is not? All they can say is, that they have no evidence that it 23 so. Their fallacy, therefore, is that of putting what they do not know to be, for what they know not to be. Or if they trust to the difficulty of finding anything in the case of straight lines by which the variation of the angle could have been regulated, How do they know, for example, that nature, instead of making the angle c=2R-(A+B), has not made it =2R-(A+B)+ m-P, where m the modulus is some given straight line; m m 27. M. Lacroix avows the difficulty which exists; and contents himself with simplifying the Axiom of Euclid by confining it to the case where two straight lines are intersected by a third, to which they both are perpendicular. On which he supposes it to be taken for granted, that if one of the straight lines turns inwards either way, it will cut the opposite perpendicular on the side on which it turns inwards, Which, excepting the simplification arising from taking the case where the angles are right angles, appears to be the same argument as Professor Leslie's. 28. A demonstration is offered in the Elémens de Géométrie of M. Lacroix,† and attributed to M. Bertrand, which is the hardest of any to convince of weakness, and takes the strongest hold of the difficulty which exists in distinguishing between observation and mathematical proof. It proceeds by stating, that any angle may be multiplied till it equals or exceeds a right angle. If then there be taken a right angle, and angles cut off from it equal to the supposed angle till they equal or exceed the whole, and at any distance from the angular point be drawn a perpendicular from one of the straight lines that make the right angle, and of course a parallel to the other, and other perpendiculars at the same distance in succession from one another; on all the lines being prolonged without limit, a certain number of the angular spaces will fill up or exceed the whole of the indefinite space included between the sides of the right angle, but the same number of the parallel bands, it is argued, will not. Whence, it is inferred, the sum of the angular spaces is greater than the sum of the parallel bands, and therefore one of the angular spaces greater than one of the bands; the consequence of which is, that the line making the supposed angle with the first perpendicular, will have cut the second. All references to the equality of magnitude of infinite surfaces, in respect of the parts where they are avowedly without boundaries, are intrinsically paralogisms; for it is tantamount to saying that boundaries coincide, where boundaries are none. And the only way to arrive at safe conclusions in such cases, is to demand to be shown the magnitudes asserted to be equal, in some stage where boundaries exist, and then see what can be established touching the consequences of extending the boundaries without restriction assigned. When it is affirmed that the surface of the angles is equal to or greater than the surface of the right angle, and the surface of the bands is not, to reduce this to anything reasonable and precise, it is necessary that it be understood to mean, that if circles of greater and greater radii be drawn about the angular point D, there will at some time be a portion of the quadrant exterior to the bands, greater than all the portions of bands of an altitude equal to the radius, which are exterior to the quadrant-, and further, that this will be the case, however the distance between the parallels may be increased. Now that this various senses, see Legendre's Elémens de Géométrie, 12ème édition, "Il est d'abord évident que si on ajoute un angle quelconque edh un nombre suffisant de fois à lui-même, en hdh', h'dh", hdh"", h""'dh""", on parviendra toujours à former un angle total edh""" plus grand que l'angle droit edb; mais si l'on élève sur la droite DB les perpendiculaires DE et ne saurait remplir l'angle droit ED B, quelque nombre de fois qu'elle soit FG, prolongées indéfiniment, on formera une bande indéfinie EDFG, qui ajoutée à elle-même. En effet, si l'on prend FK=DF, et qu'on élève K L perpendiculaire sur AB, que l'on plie ensuite la figure le long de FG, la bande E DFG couvrira exactement la bande GF KL; car les angles G F D, GFK, étant droits, la partie DF tombera sur FK; et comme D F F K par construction, le point D se placera sur le point K; de plus, l'angle FKL étant droit aussi bien que E D F, la ligne D E se placera sur KL. Cela posé, puisqu'on peut prendre sur la droite indéfinie DB autant qu'on voudra de parties égales à D F, sans arriver à son terme, on formera un nombre aussi grand qu'on voudra de bandes égales à E DFG, sans pouvoir couvrir l'espace indéfini compris entre les deux côtés de l'angle droit ED B. Il suit de là que, considérées relativement à leurs limites latérales, la surface de l'angle A+Bedh'est plus grande que celle de la bande EDPG. Si done on construit dans cette bande, sur la droite E D, un angle EDH égal à edh, il ne pourra demeurer contenu entre les lignes BD et FG; son côté D H coupera nécessairement la droite F G." 2 R again being equal in triangles with different angles, * to z Xwhere z shall be some grand modulus existing in nature, which (for the sake of removing the argument from vulgar experience) may be supposed to be of very great dimension, as for instance equal to the radius of the earth's orbit? If an astronomer should arise and declare he had found astronomical evidence that this was true, how would the supporters of the analytical proof proceed to put him down? And would they not find themselves in the situation of those prophets, who find it easier to prophesy after the fact, than while the result is in abeyance? + • Without some provision of this kind, the expression would present a straight line m such, that no straight lines making angles with its extremities would ever meet; which could not be, for to the extremities of any straight line others may be drawn making angles with it, from any point not in the same straight line with the first. + For reference to a number of places where this subject is agitated in lorsqu'on applique l'angle droit edb sur l'angle droit E DB, ces deux surfaces "Pour sentir la force de cette démonstration, il faut bien concevoir que doivent toujours coïncider entre leurs limites latérales de et db, D E et DB, quelque loin qu'on les prolonge : alors on verra que si les angles construits dans les bandes n'en sortaient pas, ils laisseraient un vide indéfini, après la dernière bande et un autre dans chaque bande; mais celui-ci, qui a toujours lieu près de leur sommet, est plus que compensé par les espaces qui leur deviennent communs quand ils sont sortis des bandes, parce que leurs côtés se croisant, ils se recouvrent en partie: tel est l'espace M N O, commun aux angles BDH, GFH'. Avec cette explication, il ne doit rester, à ce que je crois, aucun doute fondé sur ce que l'infini entre dans les considérations précédentes; car il ne s'agit que de concevoir qu'il est toujours possible de placer dans l'angle droit un nombre de bandes qui surpasse un nombre donné, quelque grand que soit ce dernier."--Elémens de Géométrie, par S. F. Läcroix, 13ème édition, Note, p. 23. The point really taken for granted in the formation of the conclusion above, is that the perpendiculars will at all events cut some of the straight lines that divide the right angle into smaller angles; for if this did not happen, there would be an end of the persuasion that, of the areas above mentioned, one will necessarily grow greater than the other. And that these straight lines or any of them will ever cut, is a mere taking for granted of the matter in question, viz., of Euclid's axiom that straight lines making with another straight line, angles on the same side together less than two right angles, will meet. It may be a case in which the empirical indication is very prominent, but still it is only empirical. There is no angle where some perpendicular may not be drawn from the base that shall meet the other line; for a perpendicular may always be let fall upon the base. But the question is, whether it has been geometrically proved of any angle, however small, that the perpendicular on removing to a greater distance, as for instance to the distance of the fixed stars, may not make smaller and smaller angles at the section, and at last cease to cut at all. There is no use in saying, it does not look as if it would; the question on which the bet is depending, is whether any universal reason has been pointed out, why it never can. The present, therefore, may be concluded to be another, though a very complicated and ingenious case, in which empirical inference is substituted for geometrical proof. will be true, is founded solely upon saying, "Make the experiment | The infirmity of this is, that it is taken for granted there will be by drawing on a piece of paper, and you will find it always is so." formed (m-1) triangles at M as represented, in the same manner But what the geometrician wants to know, is the principle upon as if AH KL was one straight line. Whereas it is demonstrable, which one of these areas will necessarily in all cases grow larger that A HK, HK L, &c., must all be angles less than the sum of two than the other; not to be shown the fact, that it grows larger in right angles on the side towards c G, and that before the number instances produced to him; for by the same rule he might set of points H, K, &c., at which new triangles are formed further and down that spheres are as the cubes of their diameters, on being further from A amounts to m-2, the formation of new triangles in shown that an iron ball of two inches diameter weighs eight that direction must cease, in consequence of the angles M KL, &c., times as much as another of one inch. He does not want the fact, becoming greater than two right angles on the side removed from but the reason of the fact. A. By which the intended proof falls to the ground. For since the angles of the triangle ABC are (by the Hyp.) less by a than two right angles, and the angles of the triangle HBA (by the Proposition preceding) are not greater than two right angles; the four angles of the quadrilateral figure A CBH must be less than four right angles, by at the least ; and because the angles H B C and H B E, or HBC and ACB are equal to two right angles, the remaining angles of the quadrilateral figure, A H B and HAC (and consequently A H B and K HB or the angle á H K) must be less than two right angles, by at the least ; and in like manner the angles HK L, &c. Wherefore if AH, HK, &c., be prolonged, the angles K HN, L K 0, &c., must be each equal at the least to x. And because BAC, BAM are together equal to two right angles, and BAC, ACB, ABC are (by the Hyp.) less by z than two right angles; BA M-2 must be equal to the sum of ACB and A BC; and because 4 P-mx is (by the Hyp.) less than the sum of ACB and ABC, it must be less than BAM- and 4 P-(m-1) must be less than B A м, and (m-1)x must be greater than 4 PBAM; and because BAM is less than two right angles, 4 P-BAM is greater than two right angles, still more therefore must (m-1)x be greater than two right angles, And because the angles HAC, A HB have been shown to be together less than two right angles by at the least, HAC must be less than two right angles by more than x, and still more must the angle MHA (which is less than HAC the exterior and opposite). Whence, because (m-1)≈ is greater than two right angles, and MHA is less than two right angles by more than z; MHA must be less than (m-2)x, and MHN must be greater than 2 P-(m-2), and MHK must be greater than 2 P-(m-2)x+x. And because the angle M KO is greater than MHK (for it is the exterior and opposite) it must be greater than 2 P-(m-2)x+z; and the angle KL must be greater than 2 P-(m−2) x+2x; and so on. Wherefore, before there have been taken (m-2) points as H, K, &c., the angle, as MK L, must be greater than 2P, and there must fail to be formed a new triangle on the side removed from A as required for the intended proof. 29. A demonstration, attributed to Mr. Ivory, is presented in the Notes to Professor Young's "Elements of Geometry," which, curiously enough, contains the elements of its own dissolution. It must be premised that it has previously been demonstrated (as may be done irrefragably in many ways) that the three angles of a rectilinear triangle cannot be together greater than two right angles. * ➡ The three angles of any triangle are equal to two right angles." "If what is affirmed be not true, let the three angles of the triangle AC B be less than two right angles, and let the defect from two right angles be equal to the angle z. Let P stand for a right angle, and find a multiple of the anglez, viz., mx, such that 4 P-mx, or the excess of four right angies above the multiple angle shall be less than the sum of the two angles A C3 and A BO of the proposed triangle.” "Produce the side C B, end cut off B B, EF, F G, &c., each equal to BC, so that the whole CG shall contain CB, m times; and construct the triangles BHK, EKF, FL G, &c., having their sides equal to the sides of the triangle AC B, and, consequently, their angles equal to the angles of the same triangle. In OA produced take any point м, and draw HM, KM, LM, &C.; AH, HIK, KL, &c." 30. Professor Young proposes this demonstration with an alteration, consisting in taking such a muitiple of x, that me may exceed 4 P. But if there cannot be constructed (m-2) triangles 4 P-sum of ACB and ABC; as supposed, when m is greater than The number of demonstrations proposed on the subject of Parallel Lines is evidence of the anxiety felt by geometrical writers upon the subject. If an erroneous account has been given of any cited above, the references will supply the means of correction. quadrilateral CGLM, together with 6m P-8 P, cannot exceed 6mr-8P+ 4 P-ma. Wherefore, by taking the same thing, viz., 6m P-8P, from the two unequal things, the four angles of the quadrilateral CGLM cannot exceed 4 P-MT. But 4P-mx is less than the sum of the two angles ACB and LGF: wherefore, à fortiori, the four angles of the quadrilateral cannot exceed the sum of the two angles ACB, LG F; that is, a whole cannot exceed a part of it, which is absurd. Therefore the three angles of the triangle ABC cannot be less than two right angles.” "And because the three angles of a triangle can neither be greater nor less than two right angles, they are equal to two right angles." "By help of this proposition," observes Mr. Ivory, "the defect in Euclid's Theory of Parallel Lines may be removed."-Elements of Geometry, by J. R. Young, Professor of Mathematics in Belfast College, notes, p. 178. "I shall, however, venture to suggest a trifling improvement, which the above reasoning appears to admit of, and thereby obviate an objection that might be brought against it." "It might be said, and with reason, that we have no right to assume that, in every case, a multiple of may be taken, such that 4P-ma may be less than the sum of the two angles ACB and ABC; for these angles may be so small that their sum shall be much less than the angle z, however small this be assumed; and although ir-mz must also be less than z, it may nevertheless be comparatively much greater than the sum of the angles ACB, ABC; in which case the above conclusion cannot be drawn." "1e angles of all the triangles into which the quadrilateral figure CGIM is divided, constitute the four angles of that figure, together with the angles round each of the points u, K, &c., and the angles directed into the interior of the figure, at the points A, B, E, F, &c. But all the angles roun the points II, K, &c., of which points the number is m-2, are equal to (-2)4P, or to 4mP-8r; and all the angles at the points A, B, E, F, &c, are equal to m times 2 P. Wherefore the sum of all the angles of all the triangles into which the quadrilateral OGLM is divided, is equal to the four angles of that figure, together with 4 m P-8P+2mr=6m r—8 r.” "Again: the three angles of the triangle ABC are, by hypothesis, equal to 2P-; and, as the number of the triangles CAB, BHE, EKP, FLG, is equal to m, tue sum of all the angles of all these triangles will be equal to "It appears, therefore, preferable to assume the multiple of a, such that 2 mr-mr. Upon each of the lines AB, HK, KL, there stand two tri- mr may exceed 4P, which is unquestionably allowable: then the subsequent angles, one above, and one below; and, as the three angles of a triangle reasoning may remain the same till we come to the inference, that the four cannot exceed two right angles, it follows that all the angles of those triangles of the quadrilateral, together with 6m P-8 P, cannot exceed 6m Pangles, the number of which is equal to 2 m-2, cannot exceed 4m P-4P. 8r+4r-mx, which obviously involves an absurdity, because 6m P-8 P Wherefore the sum of all the angles of all the triangles into which the alone exceeds 6m P-8P+4P-m; since this latter expression results from quadrilateral CGLM is divided cannot exceed 4mP-4+2mr-mx=6m Padding to the former a less magnitude, viz., 4P, and taking away a greater --8P+4P-ME-" viz., mz, for by hypothesis 4mx."—Elements of Geometry, by J. R. Young, p. 179. "It follo Te from what has now been proved, that the four angles of the M The following is the passage referred to, in p. 313, and we | c, and with sides m', n',p', respectively opposite to them. Since insert it because of its great ingenuity and value ::A and B are not changed, we shall still, in this new triangle, have By superposition, it can be shown immediately, and without m'p v: (A, B), and n' =p': (A, B). Hence m: m' =n;n=p: any preliminary propositions, that two triangles are equal when P. Hence, in equiangular triangles, the sides opposite the equal they have two angles and an interjacent side in each equal. Let angles are proportional. us call this side p, the two adjacent angles A and B, the third The proposition concerning the square of the hypotenuse is a angle c. This third angle c, therefore, is entirely determined, consequence of that concerning equiangular triangles. Here when the angles A and B, with the side p, are known; for if then are three fundamental propositions of geometry,—that conseveral different angles c might correspond to the three given mag-cerning the three angles of a triangle, that concerning equiangular nitudes A, B, P, there would be several different triangles, each triangles, and that concerning the square of the hypotenuse, having two angles, and the interjacent side equal, which is impossi- which may be very simply and directly deduced from the ble; hence the angle c must be a determinate function ofthe three consideration of functions. In the same way, the propositions quantities A, B, P, which we shall express thus, C=: (A, B, P). relating to similar figures and similar solids may be demonstrated with great ease. Let the right angle be equal to unity, then the angles A, B, C will be numbers included between 0 and 2; and since c=: (A, B, p), the line p cannot enter into the function . For we have already seen that c must be entirely determined by the given quantities A, B, p alone, without any other line or angle whatever. But the line p is heterogeneous with the numbers A, B, C; and if there existed any equation between A, B, C, P, the value of p might be found from it in terms of A, B, c; whence it would follow, that p is equal to a number; which is absurd: hence p cannot enter into the function 4, and we have simply c=: (A, B).* This formula already proves, that if two angles of one triangle are equal to two angles of another, the third angle of the former must also be equal to the third of the latter; and this granted, it is easy to arrive at the theorem we have in view. First, let A B C be a triangle right-angled at A; from the point A draw A D perpendicular to the hypotenuse. The angles B and D of the triangle A B D are equal to the angles B and A of the triangle B AC; hence, from what has B. just been proved, the third angle B A D is equal A D C to the third c. For a like reason, the angle D A CB, hence BAD +DAC, or BAC B+C; but the angle B A C is right; hence the two acute angles of a right-angled triangle are together equal to a right angle. Now, let B A C be any triangle, and B C a side of it not less than either of the other sides; if from the opposite angle a the perpendicular A D is let fall on B C, this perpendicular will fall within the triangle ABC, and divide it into two right-angled triangles BAD, DA C. But in the right angled triangle BA D, the two angles BAD, ABD are together equal to a right angle; in the right-angled triangle D A C, the two DAC, A C D are also equal to a right angle; hence all the four taken together, or, which amounts to the same thing, all three, B AC, ABC, A C B, are together equal to two right angles; hence in every triangle, the sum of its three angles is equal to two right angles. It thus appears, that the theorem in question does not depend, when considered à priori, upon any series of propositions, but may be deduced immediately from the principle of homogeneity; a principle which must display itself in a relation subsisting between all quantities of whatever sort. Let us continue the investigation, and show that, from the same source, the other fundamental theorems of geometry may likewise be derived. Retaining the same denominations as above, let us further call the side opposite the angle A by the name of m, and the side opposite B by that of n. The quantity m must be entirely determined by the quantities A, B, p alone; hence m is a function of A, B, P, and is one also; so that we may put P p). But m P m m Р (A, B, is a number as well as A and B; hence the function cannot contain the line p, and we shall have simply. m Let A B C D E be any polygon. Having taken any side a B, upon AB as a base, form as many triangles A B C, A B D, &c. as there are angles o, D, E. &c. lying out of it. Put the base A B=p; let A and B represent the two angles of the triangle ABC, which are adjacent to the side A B; A' and B' the two angles of the triangle G ABD, which are adjacent to the same side AB, and so on. The figure A B C D E will be entirely K determined, if the side p with the angles A, B, A', B', A", B", &c. are known, and the number of data will in all amount to 2n-3, n being the number of the polygon's sides. This being granted, any side or line x, any how drawn in the polygon, and from the data alone which serve to determine this polygon, will be a function of those given quantities; and since. must be a number, we may suppose F : V: (A, B, a′, B′, &c.) or x=p↓ (A, B, A', B′, &c.), and the function will not contain p. If with the same angles, and another side p', a second polygon be formed, the line ' corresponding or homologous to x will have for its value ′ = p v': (A, B, A', B', &c.); hence :p: p'. Figures thus constructed might be defined as similar figures; hence in similar figures the homologous lines are proportional. Thus, not only the homologous sides and the homologous diagonals, but also any lines terminating the same way in the two figures, are to each other as any other two homologous lines whatever. S Let us name the surface of the first polygon S; that surface is homogeneous with the square p2; hence must be a number, p containing nothing but the angles A, B, A', B', &c.; so that we shall have S = p2 : (A, B, A', B', &c.); for the same reason, S being the surface of the second polygon, we shall have S' = p2 ¢ ; (A, B, A', B', &c.) Hence S: S'p: p"; hence the surfaces of similar figures are to each other as the squares of their homolo gous sides. Let us now proceed to polyedrons. We may take it for granted, that a face is determined by means of a given side p, and of the several given angles A, B, C, &c. Next, the vertices of the solid angles which lie out of this face, will be determined each by means of three given quantities, which may be regarded as so many angles; so that the whole determination of the polyedron depends on one side, p, and several angles A, B, C, &c., the number of which varies according to the nature of the polyedron. This being granted, a line which joins to no vertices, or more generally, any line a drawn in a determinate manner in the polyedron, and from the data alone which serve to construct it, will be a function of the given quantities P, A, B, C, &c.; and since must be a number, the function equal to will contain Р (A, B, C, &c.) The surface of the solid is homogeneous to p2; nothing but the angles A, B, C, &c., and we may put x=p: hence that surface may be represented by på: (A, B, C, &c.) : its solidity is homogeneous with p3, and may be represented by p dent of p. п: (A, B, C, &c.), the functions designated by v, and п being inde Р B), orm=p: (A, B). Hence, also, in like manner, n = p ↓ (B, A) gainst this demonstration it has been objected, that if it were applied word for word to spherical triangles, we should find that two angles being known, are sufficient to determine the third, which is not the case in that species of triangles. The answer is, that in spherisal triangles there exists one element more than in plane triangles, the radius of the sphere, namely, which must not be omitted in our reasoning. Let r be the radius; instead of c A, B, P) we shall now have c= (A, B, P, r), or by the law of homo eneity, simply c = 4| But since the ratio is a number as we 1 a3 A, B, C, there is nothing to hinder from entering the function ♦, an consequently, we have no right to infer from it, that c❖ (A, B). Suppose a second solid to be formed with the same angle A, B, C, &c., and a side p' different from p; and that the solids so formed are called similar solids. The line which in the former solid was po (A, B, C, &c.), or simply po, will in this new solid ne come p'; the surface which was pay in the one, will now become p' in the other; and, lastly, the solidity which was p3n in th will now become p's n in the other. Hence, first, in similar solids, the homologous lines are proportional; secondly, their surfaces are as the squares of the homologous sides; thirdly, their solidities are as the cubes of those same sides. be the circumference, and s the surface of the circle whose radius from, is put in the genitive: as, die meisten Verlufte sind eines is r; since there cannot be two unequal circles with the same Grsages fähig, most losses are capable of reparation; die Erre ist radius, the quantities and must be determinate functions of x voll ber Güte des Herrn, the earth is full of the goodness of the Lord. tion. skilled. Gewöhnt, used to; in the habit. nate functions of r and A; hence and are also similar func-Gewiß, sure; certain. I до : A. Hence : x =r:r', and y: y = r2: r2; hence similar arcs, or the arcs of similar sectors, are to each other as their radii; and the sectors themselves are as the squares of the radii. By the same method we could evidently show, that spheres are as the cubes of their radii. = In all this we have supposed that surfaces are measured by the product of two lines, and solids by the product of three; a truth which is easy to demonstrate by analysis, in like manner. Let us examine a rectangle, whose sides are p and q; its surface, which must be a function of p and q, we shall represent by: (P, q). If we examine another rectangle, whose dimensions are p+p' and q, this rectangle is evidently composed of two others; of one having p and q for its dimensions, of another having p' and q; so that we may put : (p + P′, q) = p : (P, q) + 4 : (P′, q). Let p'p; we shall have (2 p, q) = 2 + (P, q). Let p 2 p; we shall have (3 p, 1) = 4 (P, 1) + ¢ (2 P, q) = 34 (p, q). Let p'=3p; we shall have (4 p, q) = ¢ (P, q) + ¢ (3 p, q) = 4 ¢ (P, q). Hence generally, if k is any whole number, we shall have ø (p. q) – 4 (k P, L); from which it fol(k p, q) = k (P, q) or kp lows that (P. ¶) is such a function of p as not to be changed by substituting in place of p any multiple of it k p. Hence this function is independent of p, and cannot include any thing except q. But for the same reason (P, q) must be independent of 9; hence (P, ) includes neither p nor q, and must therefore be = Pq limited to a constant quantity a. Hence we shall have (p,q) = apq; and as there is nothing to prevent us from taking a = we shall have (p,q)=pq; thus the surface of a rectangle is equal to the product of its two dimensions. In the very same manner, we could show, that the solidity of a right-angled parallelopipedon, whose dimensions are p, q, r, is equal to the product p q r of its three dimensions. We may observe, in conclusion, that the doctrine of functions, which thus affords a very simple demonstration of the fundamental propositions of geometry, has already been employed with success in demonstrating the fundamental principles of Mechanics. See the Memoirs of Turin, vol. ii. LESSONS IN GERMAN.-No. LXXXIII. A NOUN limiting the application of an adjective, when in Loos, free; rid. Mächtig, having; in possession. OBSERVATIONS. (3) A right regard to the observation made above, namely, (1) The genitive is in like manner put after the following im. that the dative merely marks that person or thing in reference personals: Nouns denoting the time, place, manner, intent or cause of an action, are often put absolutely in the genitive and treated as adverbs: as, Des Morgens gehe ich aus, in the morning I go out. (1) This_adverbial use of the genitive is quite common in German. In order, however, to express the particular point, or the duration of time, the accusative is generally employed, or preposition with its proper case; as, Ich werde nächsten Montag aus der Stadt gehen, I shall go out of town next Monday. $ 129. RULE. a A noun or pronoun used to represent the object in reference to which an action is done or directed, is put in the dative: as, Ich danke dir, I thank (or am thankful to) you. Er gefällt vielen Leuten, he pleases many people. OBSERVATIONS. to which an action is performed, will serve, also, to explain all (4) The rule comprehends all such verbs as the following: (5) This rule, also, comprehends all reflexive verbs that govern the dative: as, ich maße mir keinen Titel an, welchen ich nicht habe, I claim to myself no title which I have not; as, also, impersonals requiring the dative: as, es beliebt mir, it pleases me, or I am pleased; es mangelt mir, it is wanting to me, or I am wanting, &c. (6) The dative is also often used after passive verbs: as, ihnen wurde wiederstanden, it was resisted to them, i. e. they were resisted: von Geistern wird der Weg dazu beschüßt, the way thereto is guarded by angels; ihm wird gelohnt, (literally) it is rewarded to him, i. e. he is rewarded. $130. RULE. Many compound verbs, particularly those compounded with er, ver, ent, an, ab, auf, bei, nach, vor, zu and wider, require after them the dative; as, Ich habe ihm Geld angeboten, I have offered him money. S 131. RULE. An adjective used to limit the application of a noun, where in Sei deinem Herrn getreu, be faithful to your master. to us. OBSERVATIONS. (1) Under this rule are embraced (among others) the following adjectives: ähnlich, like; angemessen, appropriate; angenehm, agreeable; anstößig, offensive; beannt, known; beschieben, destined; eigen, peculiar; fremd, foreign; gemäß, according to; gemein, comtich, agreeable; nahe, near; überlegen, superior; willkommen, welmon; gewachsen, competent; gnädig, gracious; heilsam, healthful; come; widrig, adverse; dienstbar, serviceable; gehorsam, obedient ; nüglich, useful. § 132. RULE. A noun or pronoun which is the immediate object of an active transitive verb, is put in the accusative: Wir lieben unsere Freunde, we love our friends. Der Hund bewacht das Haus, the dog guards the house. (1) The accusative, as before said, being the case of the direct (1) The dative is the case employed to denote the person or or immediate object (§ 129. 1.), is used with all verbs, whatever the thing in relation to which the subject of the verb is repre- their classification in other respects, that have a transitive sigsented as acting. Compared with the accusative, it is the case nification. Accordingly, under this rule come all those imperof the remote object: the accusative being the case of the im-sonal and reflexive verbs that take after them the accusative; mediate object. Thus, in the example, ich schrieb meinem Vater einen Brief, I wrote (to) my father a letter, the immediate object is a letter; while father, the person to whom I wrote, is the remote object. The number of verbs thus taking the accusative with the dative, is large. (2) On the principle explained in the preceding observation may be resolved such cases as the following: es thut mir leid, it causes me sorrow, or I am sorry, es wird mir im Herzen wehe thun, it will cause pain to me in the heart (it will pain me to the heart). all those verbs having a causative signification, as, fällen, to fell, (2) Lehren, to teach; nennen, to name; heißen, to call; scheiten, to |