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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 c, 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 2R-P tang. of semi-difference of the angles at the

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 Yi.at quantities must be ejected to preserve the law of homogeneity, is visionary till it is known what quantities may or may not subsequently 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 of the admission of R), is necessarily non-existent in their own. Take, for example, the hyperbolic triangle A H G, towards the end of 16 of this collection. in this it is plain that if the line A B and the straight lines A G and G H 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 mature had contrived that the three angles of a triangle should not be always equal to two right angles, the proportionality of the sides of silhijar Uriangles 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 propósed demonstration, do they know that it is not ? All they can say is, that they have no evidence that it ts 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, .

bow do they know, for example, that nature, instead of making

the angle c = 2 R-(A + F), has not made it =2n - (A-F B)+

**, where on the modulus is some given straight line; m 472,

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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,t ańd 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 opposed angle with the first perpendicular, will have cut the SCCOI) Ol. &

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, 122me 6dition,
AVotec, p. 287.
* Elémens de Géométrie, par S. F. Lacroix. 13&me édition, p. 22.
* “On doit cependant excepter de ce reproche la démonstration donnée
par M. Bertrand; elle me paru la plus simple et la plus ingénieuse de toutes
celles que je connais; en voici le fond.”
“Il est d'abord €vident que si on ajoute un angle quelconque ed h un
nombre suffisant de fois à lui-même, en hdh', h’dh”, h”dh”, h’’’dh”, on
parviendra toujours à former un angle total edh” plus grand que l'angle
droit edb; mals si l’on élève sur la droite D B les perpendiculaires D E et
FG, prolongées indéfiniment, on formera une bande indéfinie E D F G, qui
ne saurait remplir Pangle droit E D B, quelque nombre de fois qu’elle soit
ajoutée à elle-même. En effet, si l’on prend F K=DF, et qu'on €lève K L.
perpendiculaire sur, AB, que l’on plie eusuite la figure le long de F G, la
bande B D F G couvrira exactement la bande G F K L; car les angles G F D,
G F K, 6tant droits, la partie D F tombera sur FK; et comme D F=F K par
construction, le point D se placera sur le point K ; de plus, l'angle FRL
étant droit aussi bien que E D F, la ligne D E se placera sur R L. Cela posé,
puisqu'on peut prendre sur la droite indéfinie D B autant qu'on voudra de
parties égales à D F, sans arriver a son terme, on formera un nombre aussi
grand qu'on voudra de bandes égales à E D FG, sans pouvoir couvrir l'espace
indéfini compris entre les deux cótés de l'angle droit E D B, Il suit de la
que, considérées relativement à leurs limites latérales, la surface de l'angle
edh est plus grande que celle de la bande B D F G. Si donc on construit
dans cette bande, sur la droite R D, un angle E D H &gal a edh, il ne pourra
demeurer contenu entre les lignes E D et F g; son côté D H coupera néces-
sairement la droite F G.”
“Pour sentirla force de gette démonstration, il faut bien concevoir que
lorsqu'on applique l'angle droit edb sur l'angle droit R D B, ces deux surfaces
doivent toujours coincider entre leurs limites latérales de et db, D E et D B,
quelque loin qu'on les prolonge: alors on verra que siles angles construits
dans les bandes n'en sortaient pas, ils laisseraient un vide indéfini, après la
dernière bande et unautre dams 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 cótés
se croisant, ils se recouvrent en partie: tel est l'espace M. No, communaux
angles E D H, G F H'. Ayec cette explication, il me doit rester, a ce que je
crois, aucundoute fondé sur ce que l'infini entre dans les considérations
précèdented 3 car il me 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. La.
croia, 132me édition, Note, p. 23.

will be true, is founded solely upon saying, “Make the experiment by drawing on a piece of paper, and you will find it always is so.” but what the geometrician wants to know, is the principle upon which one of these areas will necessarily in all cases grow larger than the other; not to be shown the fact, that *...*. larger in instances produced to him; for by the same rule, he might set down that spheres are as the cubes of their diameters, on being shown that an iron ball of two inches diameter weighs eight times as much as another of one inch. He does not want the fact, but the reason of the fact. 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, flowever small, that the perpendicular on removing to a greater distance, as for instance to the distance of the fixed stars, may isot 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. 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 A G B be less than two right angles, and let the defect from two right angles be equal to the anglez. Let P stand for a right angle, and find a multiple of the angle ac, viz., mac, such that 4 P-ma, or the excess of four right angles above the multiple angle shall be less than the sum of the two angles A Co and A B C of the proposed triangle.”

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“Produce the side c B, end cut off B e, E F, FG, &c., each equal to B C, so that the whole c G shall contain o B, m times; and construct the triangles b H. E., R K F, F L G, &c., having their sides equal to the sides of the triangle A C B, and, consequently, their angles equal to the angles of the same triangle. In G A produced take any point M, and draw H. M., K.M., L. M., &c.; A H., H K, K L., &c.” “Al time angles of all the triangles into which the quadrilateral figure .C. G. I. M is divided, constitute the four angles of that figure, together with the angles round each of the points H, 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 H, K, &c., of which points the number is m-2, are equal to (m--2)4 e, or to 4m P-8 P; 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 c GLM is divided, is equal to the four angles of that figure, together with 4 m. P-8 P--2m P=6m P-8 P.” “Asgain: the three angles of the triangle ABC are, by hypothesis, equal to 2 P-2'; and, as the number of the triangles CAB, BHE, B K F, FLG, is equal to m, tue sum of all the angles of all these triangles will be equal to 2 m P-ma. Upon each of the lines AB, HK, KL, there stand two triangles, one above, and one below ; and, as the three angles of a triangle cannot exceed two right angles, it follows that all the angles of those triangles, the number of which is equal to 2 m—2, cannot exceed 4th P-4 P. Wherefore the sum of all the angles of all the triangles into which the quadrilateral CG LM is divided cannot exceed 4m P-4P+2 mr-mat-6 m P --8 e-o-4 r—otz-” “It follo's from what has now been proved, that the four angles of the

The infirmity of this is, that it is taken for granted there will be formed (m—1) triangles at M as represented, in the same manner as if A H K L was one straight line. Whereas it is demonstrable, that A. H. K., H K L., &c., must all be angles less than the sum of two right angles on the side towards c G, and that before the number of points H, K, &c., at which new triangles are formed further and further from A amounts to m—2, the formation of new triangles in that direction must cease, in consequence of the angles M. KL, &c., becoming greater than two right angles on the side removed from

.A. By which the intended proof falls to the ground.

For since the angles of the triangle A B C 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 cis H. must be less than four right angles, by at the least 2; and because the angles H B C and H B E, or H B C and A ch are equal to two right angles, the remaining angles of the quadrilateral figure, A H B and H A G (and consequently A H B and K H B or the angle A. H K) must be less than two right angles, by at the least a ; and in like manner the angles H. K. L., &c. herefore if A H., H K, &c., be prolonged, the angles K. H. N., L Ko, &c., must be each equal at the least to a. And because B A C, B A M are together equal to two right angles, and B A C, A C B, A B C are (by the Hyp.) less by a than two right angles; BA M-2 must be equal to the sum of A ch and A B C ; and because 4 P-ma is (by the Hyp.) less than the sum of A c B and A B C, it must be less than B AM-2 and 4 P-(m—1): must be less than B A M, and (m—1)a must be greater than 4 PB AM; and because B AM is less than two right angles, 4 P-EAM: is greater than two right angles, still more therefore must (m—1)2 be greater than two right angles. And because the angles H. Ac, A H B have been shown to be together less than two right angles by at the least z, H. Ac must be less than two right angles by more than ar, and still more must the angle M H A (which is less than H. A c the exterior and opposite). Whence, because (m—1)2 is greater than two right angles, and M H A is less than two right angles by more than 2; M. H. A must be less than (m—2)2, and M. H. N. must be greater than 2 P-(m—2) ar, and M H K must be greater than 2 P-(m—2)2+2. And because the angle M. Ko is greater than M H K (for it is the exterior and opposite) it must be greater than 2 P-(m—2)2+2; and the angle M KI, must be greater than 2 P-(m—2)2+2+; and so on. W.: before there have been taken (m—2) points as H, K, &c., the angle, as MRI, must be greater than 2 P, and there must fail to be formed a new triangle en the side removed from A as required for the intended proof. 30. Professor Young proposes this demonstration with an alteration*, consisting in taking such a muitiple of z, that ma may exceed 4 P. But if there cannot be constructed (m—2) triangles 4P-sum of Ach and A R c; as supposed, when m is greater than

3: 4 P. still more cannot (m—2) be constructed when m is greater than —

3C 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 c GLM, together with 6m P-8 P. cannot exceed 6m P—8F-14 P-ma. Wherefore, by taking the same thing, viz., 6m P-8F, from the two unequal things, the four angles of the quadrilateral G GLM cannot exceed 4?—ma. But 4r—ma is less than the sum of the two angles A cop and L G F : wherefore, a fortiori, the four angles of the quadrilateral cannot exceed the sum of the two angles ACB, LGF; that is, a whole cannot exceed a part of it, which is absurd. Therefore the three angles of the triangle A B C 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 defectin Euclid's Theory of Parallel Lines may be removed.”—Elements of Geometry, by J. R. Young, Professor of Mathematies 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 salā, and with reason, that we have no right to assume that, in every case, a multiple of a may be taken, such that 4P-ma: may be less than the sum of the two angles Acts and AB C ; for these angles may be so small that their sum shall be much less than the angle *, however small this be assumed; and although 4p–ma: must also be less than ar, it may nevertheless be comparatively much greater than the sum of the angles Ace, ABC; in which case the above conclusion cannot be drawn.” “it appears, therefore, preferable to assume the multiple of z, such that ma: may exceed 4P, which is unquestionably allowable : then the subsequent reasoning may remain the same till we come to the inference, that the four angles of the quadrilateral, together with 6m P, cannot exceed 6m Pgoor-mo, which obviously involves an absurdity, because 6 on F-8F alone exceeds 6m P-8 P-H4P-ma;; since this latter expression results from adding to the former a less magnitude, viz., 4 P, and taking away #97 or viz., n., for by hypothesis 4.5-m2.”-Elements of Geometry, by J. R.

Young, p. 179.

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The following is the passage referred to, in p. 313, and we insert it because of its great ingenuity and value:—

By superposition, it can be shown immediately, and without any preliminary propositions, that two triangles are equal when they have two angles and an interjacent side in each equal. Let aus gall this side p, she two adjacent angles A and B, the third angle c. This third angle c, therefore, is entirely determined, when the angles A and B, with the side p, are known; for if several different angles C might correspond to the three given magnitudes A, B, p, there would be several different triangles, eac having two angles, and the interjacent side equal, which is impossible; hence the angle c must be a determinate function ofthe three quantities A, B, p, which we shall express thus, G = @: (A, B, p). 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 = ?: #. B, p), the line à cannot enter into the function p. For we ave already seen that c must be entirely determined by the given $o A, B, p alone, without any other line or angle whatever. ut the line p is heterogeneous with the numbers A, B, C ; and if there existed any equation between A, B, C, # the value of 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 p, and we have . 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 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 A C ; hence, from what has B C just been proved, the third angle B A D is equal D to the third C. For a like reason, the angle D A c = B, hence *AD+D AC, or B.A. C= 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 Athe perpendicular A D is let fall on B C, this perpendicular .# fall within the triangle A B C, and divide it into two right-angled triangles BA D, D.A. C. But in the right-angled triangle B. A D, the two angles B A P, A B D are together equal to a right angle; in the right-angled triangle D.A.G., the two D A c, A c is are also equal to a right angle,; hence all the four taken together, or, which amounts to the same thing, all three, B A c, A B c, A C B, are together equal to two right angles; hence in every triangle, the swim of its three angles is equal to two right angles. It thus appears, that the theorem in question does not depend, when considered a priori, upon any series of propositions, but may be deduced immediately from the principle of homogenéity; 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 dérived. 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 on is a function

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* 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 know n, 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 = p(A, B, p, r), or by the law of

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C, and with sides m', 'n',p', respectively opposite to them. Since A and B are not changed, we shall still, in this new triangle, have 72° TÉ. V : (A, B), and n = p \, : (A, B). Hence m: on' = non' = p : p'. Hence, in equiangular triangles, the sides opposite the equal *::::: are proportional. he proposition concerning the square of the hypotenuse is a

consequence of that concerning equiangular triangles. Here then are three fundamental propositions of geometry, that concerning the three angles of a triangle, that concerning equiangular o and that concerning the square of the Fo which may be very simply and directly deduced from the consideration of functions. In the same way, the propositions relating to similar figures and similar solids may be demonstrated with great ease.

Let A B C D E be any o; Having taken any side A B, upon A B as a base, form as many triangles A B C, A B D, &c. as there are angles C, D, E. &c. lying out of it. Put the base A B = #: let A and B represent the two angles of the triangle A B C, which are adjacent to the side A B ; A' and B’ the two angles of the triangle G A B D, which are adjacent to the same side A B, 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 F data will in all amount to 2 m —-3, n being the number of the polygon's sides. This being granted, any side or line w, any how drawn in the polygon, and from the data alone which serve to determine this polygon, will be a function of those

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p function V, will not contain p. If with the same angles, and another side p" a second polygon be formed, the line a-' corresponding or homologous to a will have for its value a = p V': o B, A, B', &c.); hence a 2' = 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. Let us name the surface of the first polygon S ; that surface is

homogeneous with the square p’; hence; must be a number; p

containing nothing but the angles A, B, A, B', &c.; so that we shall have S = p” p : (A, B, A, B', &c.); for the same reason, S^ being the surface of the second polygon, we shall have S' = p^* @: (A, B, A', B', &c.) Hence S : S' = p^: #. ; hence the swrfaces of similar figures are to each other as the squares of their homologows 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 `...; 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 p

p nothing but the angles A, B, C, &c., and we may put a = p * : (A, B, C, &c.) The surface of the solid is homogeneous to po; hence that §urface may be represented by pov, : (A, B, C, &c.) : its solidity is homogeneous with p", and may be represented by : &B, C, &c.), the functions designated by l, and II being indeent of p. 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 fo imed are called similar solids. The line which in the former solid was p q (A, B, C, &c.), or simply p *, will in this new solid be come * @ ; the surface which was p” W. in the one, will now becom ep” V, in the other; and, lastly, the solidity which was port in th one, will now become p’s m in the other. Hence, first, in similar solids, the homologous lines are proportional ; secondlw. their surfaces are as the squares of the homologous sides ; third sy, their solidities are as the cubes of those same sides. The same principles are easily applicable to the circle. Let c

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= r ; r", and y : y' = r^ : r"; 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. 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 4 (p, q). If we examine another rectangle, whose dimensions are p-H p’ and q, this rectangle is evidently composed of two others : of one having p and g for its dimensions, of another having p’ and g; so that we may put b : (p + p", q) = p : (p, q) + b : (p, q). Let p’= p ; we shall have @ (2 p, q) = 24 (p,q). Let p = 2 p.; we shall have 4 (3 p, q) = ? (p, q) + p (2 p, q) = 3 p (p, q). Let p'=3p; we shall have @ (4 p, q) = 4 (p, q) + p (3 p, q) = 4 p. (p, q). Hence generally, if k is any whole number, we shall have

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limited to a constant quantity a. Hence we shall have @ (p, o st Q, # and as there is nothing to prevent us from taking a =T, we shii have q (p,q) = p q ; 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. e may observe, in conclusion, that the doctrine of functions, which thus affords a very simple demonstration of the fundamen. tal propositions of geometry, has already been employed with success in demonstrating the fundamental principles of Mechanics. See the Memoirs of Twrin, vol. ii.

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OBSERVATIONs.

28tbürfeit, begesiren, braudjem, entbeştem, ermášuen, genießen. pflegen. jöomen, verseoleit, vergeisen, mustmesniem, moaten and martem, take more frequently, in common conversation, the accusatise. Qld)tem, Šatten and matten are more commonly construed with qui, and , (adjen, spotten and putten with sić er, before an accusative.

§ 126. RULE.

The following reflexive verbs take, in addition to the pronoun peculiar to them, a word of limitation in the genitive:

&id) alimašen, to claim. ©id erfreden, to presume. ,, annessmen, to engage in. o, crimmern, to rememoer.

, bebiencil, to use. , trfüßmen, to venture, , befleißen, to attend to. ,, eripejren, to resist. , befleißigen, to apply to. , freuen, to rejoice.

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getristen, to hope for. rismen, to boast. föinen, to be ashamed. iifferijeben, to be haughty. unterfangen, to undertake. untermeinben, to undertake. werinessen, to presume, verseşen, to be aware. peorem, to resist. weigern, to refuse. impuntern, to wonder.

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(1) The dative is the case employed to denote the person or the thing in relation to which the subject of the verb is represented as acting. Compared with the accusative, it is the case of the remote object: the accusative being the case of the immediate object. Thus, in the example, its sérieff meinem $8ater cinem 3ries, 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 : e6 tout mit seib, it causes me sorrow, or I am sorry, e3 miro mir im Şergen messe toun, it will cause pain to me in the heart (it will pain me to the heart).

calls him his deliverer.

(3) A right regard to the observation made above, namely, that the dative merely marks that person or thing in reference to which an action is performed, will serve, also, to explain all such examples as these: Şūmen bebtutet bitsts Spfer midté, to you (i.e. so far as you are concerned) this sacrifice means nothing; bie $ffrånen, bie (§urem &treit geñoften, the tears which have flowed in relation to (i. e. from) your dispute; mit tobtete ein Göuff bag $fett, a shot killed a horse for me, i.e. killed my horse; false, mit midt, Joseiner, fall not for me, little one. In such instances as the last two, the dative is often omitted in translating. (4) The rule comprehends all such verbs as the following: antiporten, to answer; banfen, to thank; bienen, to serve; broßen, to threaten ; fefflen, to fall short; fluđen, to curse; fosgen, to follow ; fränen, to do homage; gebäären, to be due ; gesassen, to please; geşöten, to pertain to; geogrójen, to obey; genigen, to satisfy; ge. reiden, to be adequate; gleijen, to resemble; jeffem, to help, &c. (5) This rule, also, comprehends all reflexive verbs that govern the dative: as, id maše mit feinen £ites am, petjen id, midt §abe, I claim to myself no title which I have not; as, also, impersonals requiring the dative: as, e6 besiest mir, it pleases me, or I am pleased; e.8 mangest mit, it is wanting to me, or I am wanting, &c. (6) The dative is also often used after passive verbs: as, is ten murbe mieberstanben, it was resisted to them, i. e. they were resisted : von Geistern mirb bet goeg bagu befoššt, the way thereto is guarded by angels; iim with gesoffnt, (literally) it is rewarded to him, i. e. he is rewarded.

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(1) The accusative, as before said, being the case of the direct or immediate object (§ 129. 1.), is used with all verbs, whatever their classification in other respects, that have a transitive signification. Accordingly, under this rule come all those impersonal and reflexive verbs that take after them the accusative; all those verbs having a causative signification, as, fillen, to fell, i. e. to cause to fall; as also nearly all verbs compounded with the prefix 6e. The exceptions are, begegnen, beffagen, befteåen, bejarren and bemoadsen.

(2) georem, to teach; mennen, to mame; jeißen, to call; sóetten, to reproach (with vile names); taufen, to baptize (christen); take after them two accusatives: as, et señrt mid bie beutsche &prade, he teaches me the German language; et memnt ion stinen Öettet, he See Sect. 53

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