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:: tang of 2 R-P

must be employed as elements in its actual calculation. These 27. M. Lacroix avows the difficulty which exists; and contents two sets are not necessarily thic same, either in number or kind. himself with simplifying the Axiom of Euclid by confining it to Take, for example, Professor Leslie's case, where c=”: (a, b, p). the case where two straight lines are intersected by a third, to

It is quite true that when a, b, p are fixed, c is fixed. But pro- which they both are perpendicular. On which he supposes it to ceed to the actual calculation of c, and very different things be taken for granted, that if one of the straight lines turns inwards appear upon the scene. For the value of c has to be collected either way, it will cut the opposite perpendicular on the side on from the well-known trigonometrical formula, that a +b: a-6 which it turns inwards.* Which, excepting the simplification

arising from taking the case where the angles are right angles, : tang. of semi-difference of the angles at the appears to be the same argument as Professor Leslie's. 2

28. A demonstration is offered in the Elémens de Géométrie of M. base c. Here then, in lieu of the solitary and heterogeneous Lacroix,t and attributed to M. Bertrand, which is the hardest of angle P, start up among the practical elements of the calculation any to convince of weakness, and takes the strongest hold of the two straight lines, in the shape of the tangents of two arcs, which difficulty which exists in distinguishing between observation and of course do not afterwards fail to conduct themselves with mathematical proof. It proceeds by stating, that any augle may perfect submission to the law of homogeneity. And with all be multiplied till it equals or exceeds a right angle. Ir then there this the proposers of the analytical proof aro bound to make be taken a right angle, and angles cut off from it equal to the their argument square; for the concession of their own demands supposed angle till they equal or exceed the whole, and at any ends in establishing the rosults of vulgar trigonometry, and not distance from the angular point be drawn a perpendicular from in altering them. On the whole, therefore, the pretence of know- one of the straight lines that make the right angle, and of course ing yiat quantities must be ejected to preserve the law of homo a parallel to the other, and other perpendiculars at the same geneity, is visionary till it is known what quantities may or may distance in succession from one another; on all the lines being not sul sequently appear among the practical elements of the prolonged without limit, a certain number of the angular spaces calculation; which is impossible in the preliminary stage. will fill up or exceed the whole of the indefinite space included

The point, then, which the supporters of the analytical proof between the sides of the right angle, but the same number of the will be called on to establish, is why the possibility of the appa- parallel bands, it is argued, will not. Whence, it is inferred, the rition of new elements which is visible in other cases (and which sum of the angular spaces is greater than the sum of the parallel in Professor Leslie's case they actually claim by demanding the bands, and therefore one of the angular spaces greater than oue admission of R), is necessarily non-existent in their own. Take, of the bands; the consequence of which is, that the line waking for example, the hyperbolic triangle A HG, towards the end of the supposed angle with the first perpendicular, will have cut the 16 of this collection. In this it is plain that if the line A B and second. the straight lines A G and on are fixed and determined, the All references to the equality of magnitude of infinite surfaces, angle au o must be one certain angle and no other. But proceed in respect of the parts where they are avowedly without boundaries, to calculate the comparative magnitude of the angle to different are intrinsically paralogisms; for it is tantamount to saying that values of A g, and there immediately start into action new elements boundaries coincide, where boundaries are none. And the only in no stinted number, viz., two constant straight lines under the way to arrive at safe conclusions in such cases, is to demand to denominations of a major and a minor axis, and a varying straight be shown the magnitudes asserted to be equal, in some stage line under the title of abscissa, to say nothing of the radius of a where boundaries exist, and then see what can be established circle and such sines or tangents of different arcs thereof as may touching the consequences of extending the boundaries without be found necessary in the process. How then do the opponents restriction assigned." When it is affirmed that the surface of the know that there are no more elements in the other case? If nature angles is equal to or greater than the surface of the right angle, had contrived that the three angles of a triangle should not be and the surface of the bands is not, to reduce this to anything always equal to two right angles, the proportionality of the sides reasonable and precise, it is necessary that it be understood to of similar triangles would not have held good, and in making mean, that if circles of greater and greater radii be drawn about Tables (for example) of the tangents to different arcs of a circle, the angular point D, there will at some time be a portion of the the magnitude of the radius of the circle must in some guise or quadrant exterior to the bands, greater than all the portions of other have been an element. The tangent of 45" to a radius of bands of an altitude equal to the radius, which are exterior to the one foot would have borne some given ratio to a foot, and the quadrant-, and further, that this will be the case, however the tangent of the same angle to a radius of two feet, instead of bear- distance between the parallels may be increased. Now that this ing 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 various senses, see Legendre's Elémens de Géométrie, 12ème édition, of the angle to a given length of radius ; in the same manner as Notes, p. 287. would be necessary if it was desired to frame a Table for finding

* Elémens de Géométrie, par S. F. Lacroix. 13ème édition, p. 22. the perpendiculars in the hyperbolic triangle for different lengths par M. Bertrand elle ine paru la plus simple et la plus Ingénieuse de toutes

+ "On doit cependant excepter de ce reproche la démonstration donnés of base. In other words, there would have been more elements. celles que je connais; en voici le fond.". That this is not so, may be a happy event; but by what evidence “Il est d'abord évident que si on ajoute un angle quelconque edh un included in their proposed demonstration, do they know that it nombre sufisant de fois à lui-même, en hur', k'da", h'dh"", "X"***, on is not? All they can say is, that they have no evidence that it parviendra toujours à former un angle total edh'"" plus grand que l'angle 23 su. Their fallacy, therefore, is that of putting what they do droit edh; mais si l'on élève sur la droite de les perpendiculaires D e et not know to be, for what they know not to be. Or if they trust ne saurait remplir l'angle droit E D B, quelque nombre de fois qu'elle soit

FG, prolongées indéfiniment, on forinera une bande indéfinie EDPG, qui to the difficulty of finding anything in the case of straight lines ajoutée à elle-même. En effet, si l'on prend Pk=DF, et qu'on élève K L by which the variation of the angle could have been regulated, perpendiculaire sur AB, que l'on plie ensuite la figure le long de po, la how do they know, for example, that nature, instead of making bande ED PO couvrira exactement la bande OP KL; car les angles at D.

GFK, étant droits, la partie de tombera sur FK; et commo DF=PK par the angle c = 2 R-(A + B), has not made it = 2 R — (A + B)+construction, le point d se placera sur le point x; de plus, l'angle PKL

étant droit aussi bien que E DP, la ligne DB se placera sur Kl. Cela posé, m—P, where m the modulus is some given straight liné; m parties égales a D , sans arriver à son terme, on formera un nombre aussi

puisqu'on peut prendre sur la droite indétinie D B autant qu'on voudra de

grand qu'on voudra de bandes égales à , sans pouvoir couvrir l'espace

2 R indéfini compris entre les deux côtés de l'angle droit E D B. Il suit de la again being equal in triangles with different angles, * to z x

que, considérées relativement à leurs limites latérales, la surface de langle

A + B | edh'est plus grande que celle de la bande E D PG. Si done on construit where shall be some grand modulus existing in nature, which dans cetie bande, sur la droite B D, un angle E dh égal à eith, il ne pourra (for the sake of removing the argument from vulgar experience) demeurer contenu entre les lignes BD et ra; son côté Du coupera nécesmay be supposed to be of very great dimension, as for instance sairement la droite r 0.". equal to the radius of the earth’s

orbit? If an astronomer

should lorsqu'on applique l'angle droit edb sur l'angle droit E D B, ces deux snrfaces

“ Pour sentir la force de cette démonstration, il faut bien concevoir que arise and declare he had found astronomical evidence that this doivent toujours coïncider entre leurs limites latérales

de et db, D 8 et DB, was true, how would the supporters of the analytical proof pro- quelque loin qu'on les prolonge : alors on verra que si les angles construits ceed to put him down? And would they not find themselves in dans les bandes n'en sortaient pas, ils laisseraieni un vide indéfini, après la the situation of those prophets, who find it easier to prophesy dernière bande et un autre dans chaque bande; mais celui-ci, qui a toujours after the fact, than while the result is in abeyance ? +

lieu près de leur soinmet, 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 uno, commun aux Without some provision of this kind, the expression would present a angles B DH, GP R'. Avec cetie explication, il ne doit rester, à ce que je straight line in such, that no straight lines making angles with its ex- crois, aucun doute fondé sur ce que l'infini entre dans les considérations tremities would ever meet; which could not be, for to the extremities of any précédentes ; car il ne s'agit que de concevoir qu'il est toujours possible de straight line others may be drawn making angles with it, from any point placer dans l'angle droit un nombre de bandes qui surpasse un nombre donne, not in tbe same straight line with the first.

quelque grand que soit ce dernier."--Élémeni de Géométrie, par $. 7. Itt + For reference to a number of places where this subject is agitated in croix, 13ème édition, Note, p. 23.


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 hy 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 Rut what the geometrician wants to know, is the principle upon as if AHK L was one straight line.' Whereas it is demonstrable, which one of these areas will necessarily in all cases grow larger that A HK, A K 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 co, and that before the number instances produced to him; for by the same rule he might set of points H, K, &c., at which ner 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 MKL, &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. The point really taken for granted in the formation of the con.

For since the angles of the triangle and are (by the Hyp.) clusion above, is that the perpendiculars will at all events cut less by æ than two right angles, and the angles of the triangle 1 BA some of the straight lines that divide the right angle into smaller (by the Proposition preceding) are not greater than two right angles; for if this did not happen, there would be an end of the angles; the four angles of the quadrilateral figure ACBU must be persuasion that, of the areas above mentioned, one will necessarily less than four right angles, by at the least &; and because the grow greater than the other. And that these straight lines or angles H B C and HBB, or HBC and Ac B are equal to two right any of them will ever cut, is a mere taking for granted of the angles, the remaining angles of the quadrilateral figure, A H B and inatter in question, viz., of Euclid's axiom that straight lines A A (and consequently A I B and K H B or the angle A u k) must making with another straight line, angles on the same side be less than two right angles, by at the least ; and in like manner together less than two right angles, will

meet. It may be a case the angles H K L, &c. Wherefore if AH, H K, &c., be prolonged, in which the empirical indication is very prominent, but still it is the angles K HN, LK 0, &c., must be each equal at the least to only empirical. There is no angle where some perpendicular may ạ. And because B A C, Bam are together equal to two right not' be drawn from the base that shall meet the other line; for a angles, and BAC, A C B, A B C are (by the Hyp.) less by x than perpendicular may always be let fall upon the base. But the two right angles; BAM—must be equal to the sum of ACB and question is, whether it has been geometrically proved of any angle, A BC; and because 4 P-mx is (by the Hyp.) less than the sum of however small, that the perpendicular on removing to a greater ACB and ABC, it must be less than BAM and 4p-(m-1). distance, as for instance to the distance of the fixed stars, may must be less than Bam, and (m-1)x must be greater than 4 Priot make smaller and smaller angles at the section, and at last BAM; and because Bam is less than two right angles, 4 P-BAM cease to cut at all. There is no use in saying, it does not look as is greater than two right angles, still more therefore must (m-1)& if it would ; the question on which the bet is depending, is whether be greater than two right angles. And because the angles 11 AC, any universal reason has been pointed out, why it never can. AH B have been shown to be together less than two right angles The present, therefore, may be concluded to be another, though a by at the least 3, HAC must be less than two right angles by very complicated and ingenious case, in which empirical inference more than x, and still more must the angle uHA (which is less in substitut for geometrical proof.

than Hac the exterior and opposite). Whence, because (m-1)ą 29. A demonstration, attributed to Mr. Ivory, is presented in is greater than two right angles, and MH A is less than two right the Notes to Professor Young's " Elements of Geometry," which, angles by more than x; MHA must be less than (m-2).x, and curiously enough, contains the elements of its own dissolution. Mun must be greater than 2 P-(m—2)«, and mak must It inust be premised that it has previously been demonstrated (as be greater than 2 P-(m-2)x+*. And because the angle MKO may be done irrefragably in many ways) that the three angles is greater than MH K (for it is the exterior and opposite) it o'á recöilinear triangle cannot be together greater than two right must be greater than 2 P-(m—2)x+a; and the angle ukl angies

must be greater than 2 P-(

m2) x+22; and so on. Wherefore,

before there have been taken (m-2) points as H, K, &c., the angle, • The three angles of any triangle are equal to two right angles.as MK L, must be greater than 2 P, and there must fail to be "}{wtat is affirmed be not true, let the three angles of the triangle Ac B formed a new triangle on the side removed from A as required for bel?ss than two right angles, and let the defect from two right angles but the intended proof. equal to the angle 1. Let P stand for a right angle, and find a multiplus 30. Professor Young proposes this demonstration with an alteråangies above the multiple angle shall be less than the sum of the two angles exceed 4 P. But if there cannot be constructed (m—2) triangles of the angle viz., me, such that 4 p-mror the excess of four right tion*, consisting in taking such a muitiple of %, that mu may A and A BO of the proposed triangle."

4P-sum of ACB and ABC; M as supposed, when m is greater than

4P still more cannot (m—2) be constructed 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.

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quadrilateral colu, together with 6m P-8P, cannot exceed 6mp—8P+ 4 P-ma. Wherefore, by taking the same thing, víz., 6m P-8 P, from tho two unequal things, the four angles of the quadrilateral colm cannot

exceed 4P-m.. But 4p-mz is less than the sum of the two angles E B

ACB and Lop: wherefore, a fortiori, the four angles of the quadrilateral

cannot exceed the sum of the iwo angles ACB, LGP; that is, a whole cannot "Produce thu side o B, vnd cut off B E, EF, FG, &c., each equal to BC, 50 exceed a part of it, which is absurd. Therefore the three angles of the that the whole ca shall contain CB, m times ; and construct the triangles triangle as c cannot be less than two right angles." DHE, EKF, PLO, &e., having their sides equal to the sides of the triangle "And because the three angles of a triangle can neither be greater nor AC B, and, consequently, their angles equal to the angles of the same tri- less than two right angles, they are equal to two right angles." angle. In o A produced take any point M, and draw u M, KM, LM, &c.; “By help of this proposition," observes Mr. Ivory, “ the defect in Euclid's AN, IK, KL, &c.”

Theory of Parallel Lines may be removed."-Elements of Geometry, by "1.o

anzles of all the triangles into which the quadrilateral figure 5. R. Young, Professor of Mathematics in Belfast College, notes, p. 178. OGIM is divided, constitute the four angles of that figure, together with • "I shall, however, venture to suggest a trifling improvement, which the argles round each of the points ul, K, &c., and the angles directed into the above reasoning appears to admit of, and thereby obviate an objection the iu erior of the figure, at the points A, B, E, , &c. But all the angles that might be brought against it." rouns Le roints 1, K, &c., of which points the number is in-2, are equal " It might be said, and with reason, that we have no right to assume that, to (m -2; 4P, or to 4m P~-8P; and all the angles at the points' A, B, E, P, in every case, a multiple of a may be taken, such that 4 P-me may be less &c, tre equal to m times 2 p. Wherefore the sum of all the angles of all the than the sum of the iwo angles A CB and ABC; for these angles may be so triangles

into which the quadrilateral OG LM is divided, is equal to the four small that their sum shall be much less than the angle z, however small angle of that figure, together with 4 m P-8P+2m P=6 in P-3 P."

this be assumed; and although 4p--me must also be less than s, it may “nkain: the three angles of the triangle ABC are, by hypothesis, equal to nevertheless be comparatively much greater than the sum of the angles 2p-*; ard, as the number of the triangles CA B, BUE, EKP, P LQ, is equal ACB, ABC; in which case the above conclusion cannot be drawn.” 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 m P-mx. Upon each of the lines AR, UK, KL, there stand two tri- mo may exceed 4 P, 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 tri- angles of the quadrilateral, together with 6m 8P, cannot exceed 6m P angles, the number of which is equal to 2m-2, cannot exceed 4mP-4p. 8r+ip-mx, whirb obviously involves an absurdity, because 6m P-8P Wherefore the sum of all the angles of all the triangles into which the alone exceeds 6m P-8P+48-m2; since this latter expression results from quadrilateral CGLM is divided cannot exceed 4mp-4P+2mp-mx=6m Padding to the former a less magnitude, viz., 4 P, and taking away a greater, :-*P+4-mr."

viz., mt, for by bypothesis 4">mr."- Elements of Geornelry, by J. R. " It follo Ts from what has now been proved, that the four angles of the Young, p. 179.

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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

p'V:(, B). Hence m: m' = n;n' = p; any preliminary propositions, that two triangles are equal when P. Hence, in équiangular triangles, the sides opposite the equai

By superposition, it can be shown immediately, and without me: (4, B), and = 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 it 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 demon

strated 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=0:

Let A BC D E be any polygon. Having taken
(A, B, p), the line p cannot enter into the function . For we any side A B, upon AB as a base, form as many
have already seen that c must be entirely determined by the given triangles A B C, A B D, &c. as there are angles c,
quantities A, B, p alone, without any other line or angle whatever. P, E. &c. lying out of it., Put the base A B=p;
But the line p is heterogeneous with the numbers A, B, C; and if let A and B represent the two angles of the
there existed any equation between A, B, C, P, the value of p trianglo ABC, which are adjacent to the side
might be found from it in terms of A,B,C; whence it would fol. A B; A' and B’ the two angles of the triangle G
low, that p is equal to a number; which is absurd: hence p can- A BV, which are adjacent to the same side A B,
not enter into the function ®, and we have simply c=$:(,B):* and so on. The figure A B C D E will be entirely K

This formula already proves, that if two angles of one triangle determined, if the side p with the angles A, B,
are equal to two angles of another, the third angle of the former A, B, A", B”, &c. are known, and the number of F
must also be equal to the third of the latter; and this granted, it data will in all amount to 2 n --3, n being the
is easy to arrive at the theorem we have in view.

number of the polygon's sides. This being granted, any side or First, let A B C be a triangle right-angled at

line x, any how drawn in the polygon, and from the data alone A; from the point a draw AD perpendicular

which serve to determine this polygon, will be a function of those to the hypotenuse. The angles B and D of the

given quantities; and since must be a number, we may suppose triangle A B D are equal to the angles B and A

р of the triangle BAC; hence, from what has B just been proved, the third angle B A D is equal

V: (A, B, A', B', &c.) or x=P (A, B, A', B', &c.), and the to the third c. For a like reason, the angle DAC=B, hence ? BAD+DAC, or BAC=B+C; but the angle BAC is right; hence function y will not contain p. If with the same angles, and another the two acute angles of a right-angled triangle are together equal side p, a second polygon be formed, the

line a' corresponding or to a right angle.

homologous to z will have for its value x' = PV': (A, B, A, B', Now, let Bac be any triangle, and B C a side of it not less than &c.); hence x:x' =p:p. Figures thus constructed might be either of the other sides; if from the opposite angle A the per- defined as similar figures; hence in similar figures the homopendicular a D is let fall on B c, this perpendicular will fall within Logous lines are proportional. Thus, not only the homologous the triangle A B C, and divide it into two right-angled triangles sides and the homologous diagonals, but also any lines terminatBAD, DAC. But in the right-angled triangle BAD, the two ing the same way in the two figures, are to each other as any angles 'BAD, A B D are together equal to a right angle; in the other two homologous lines whatever. right-angled triangle Da C, the two DAC, A C D are also equal to Let us name the surface of the first polygon S; that surface is a right angle; hence all the four taken together, or, which

S amounts to the same thing, all three, BAC, ABC, AC B, are homogeneous with the square p'; hence must be a number, together equal to two right angles; hence in every triangle, the sum of its three angles is equal to two right angles.

containing nothing but the angles A, B, A', B', &c.; so that we It thus appears that tho theorem in question does not depend, shall have S=p :,(A, B, 4, B', &c.); for the same reason, S when considered à priori, upon any series of propositions, but being the surface of the second polygon, we shall have S=p'ie : may be doduced immediately from the principle of homogeneity (A, B, 4, B', &c.) Hence $:$' = p :p's; hence the surfaces of à principle which must display itself in a relation subsisting similar figures are to each other as the squares of their homolobetween all quantities of whatever sort. Let us continue the gous sides. investigation, and show that, from the same source, the other Let us now proceed to polyedrons. We may take it for fundamental theorems of geometry may likewise be derived. granted, that a face is determined by means of a given side p,

Retaining the same denominations as above, let us further call and of the several given angles A, B, C, &c. Next, the vertices of the side opposite the angle A by the name of m, and the side the solid angles which lie out of this face, will be determined opposite B by that of n. The quantity m must be entirely each by means of three given quantities, which may be regarded determined by the quantities A, B, p alone; hence m is a function as so many angles; so that the whole determination of the polyof A, B, P, and is one also; so that we may put

edron depends on one side, P, and several angles A, B, C, &c., the = 4: (A, B, number of which varies

according to the nature of the polyedron. P

This being granted, a line which joins to no vertices, or more p). But is a number as well as A and B; hence the function generally, any line x drawn in a determinate manner in the polyр

edron, 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 cannot contain the line P, and we shall have simply = vi(A,


must be a number, the function equal to * will contain B), or m = P *:(A,B). Hence, also, in like manner, n =

P Now, let another triangle be formed with the same angles A, B, (A, B, C, &c.) The surface of the solid is homogeneous to pa:

nothing but the angles A, B, C, &c., and we may put x=P o: hence that surface may be represented by på¥: (A, B, C, &c.) : its

solidity is homogeneous with pi, and may be represented by pro gainst this demonstration it has been objected, that if it were applied dent of p.

1: (A, B, C, &c.), the functions designated by V, and 11 being indeword 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

Suppose a second solid to be formed with the same angle A, B, C, species of triangles. The answer is, that in spherisal triangles there exists &c., and a side p' different from p; and that the solids so formed one esement more than in plane triangles, the radius of the sphere, namely, are called similar solids. The line which in the former solid which must not be omitted in our reasoning. Letr be the radius; instead was po (A, B, C, &c.), or simply p¢, will in this new solid he come

PiA, B, P) we shall now have o = (A, B, P, r), or by the law of pp; the surface which was pk in the one, will now becore homo eneity, simply c=

in the other; and, lastly, the solidity which was psn in tb But since the ratio is a number will now become ps 11 in the other. Hence, first, in similar

solids, the homologous lines are proportional ; secondlv, their as wel as A, B, C, there lo nothing to hinder – from entering the function surfaces are as the squares of the homologous sides; thirty, their

solidities are as the cubes of those same sides. an consequently, we have no right to infer from it, that c = (A, B). The same principles are easily applicable to the circle Letc






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be the circumference, and the surface of the circle whose radius from, is put in the genitive: as, die meisten Verlufte find eines is r; since there cannot be two unequal circles with the same Ersages fähig, most losses are capable of reparation ; tie @rte ist radius, the quantities and must be determinate functions of poll per Güte des Herrn, the earth is full of the goodness of the

Lord. r; but as these quantities are numbers, the expression of them

OBSERVATIONS. cannot contain r; and thus we shall have =9, and = B, a,

(1) The adjectives comprehended under this rule are such as and ß being constant numbers. Let c' be the circumference, and follow : s' the surface of another circle whose radius is re'; we shall, as

Bedürftig, in want; needing. loos, five; rid before, have = a, and = B. Hence c : c'=r:r, and 8:8 Benitbigt, needing, wanting. Mächtig, having; in possession.

Müde, tired; weary. =p:gas; hence the circumferences of circles are to each other as Eingebent, mindful.

Satt, satiated; weary. their radii, and the surfaces are as the squares of those radii.

Fähig, capable ; susceptible. Let us now examine a sector whose radius is't. A being the

Schuldig, guilty ; indebted.

Froh, glad. angle at the centre, let z be the arc which terminates the sector,

Theilhaft, parlak g. and y the surface of that sector. Since the sector is entirely

Gewahr, aware.

lieberbrüssig, tired ; weary. determined when r and A are known, r and y must be determi- Gewärtig, waiting; in expecta- Verdichtig, suspici y is.

tion. Verlustig, having ivši; dorized nate functions of r and A; henco and Y are also similar func- Gewiß, suro; certain.


Gevöhnt, used to; in the habit. Voll, full. tions. But is a number, as well as %; hence those quantities Kundig, having a knowledge; Werth, worth; worthy.

skilled. Würdig, worthy. cannot contain r, and are simply functions of A ; so that we have Lebig, empty ; void.

Quitt, rid; free from,

Lerr, void. =o: A, and y

: A. Let x' and y' be the arc, and the sur. face of another sector, whose angle is A, and radius r'; we shall sative is often used: as, er ward seinen Bruder gewahr, he was aware

(2) After gewahr, gewohnt, log, müde, fatt, voll and werth, the accue call those two sectors similar : and since the angle A is the same of (the presence of ) his brother, i. e. he observed his brother in both, we shall have = 4:A, and = v : A. Hence ® : x

$ 125. RULE. =r:r, and y: yo = go? : que; hence similar arcs, or the arcs of similar sectors, are to each other as their radii; and the sector's

A noun limiting the application of any of the following verbs themselves are as the squares of the radii. By the same method we could evidently show, that spheres are

is put in the genitive: as the cubes of their radii.

Harren, to wait. In all this we have supposed that surfaces are measured by the Mchten, to mind, or regard. product of two lines, and solids by the product of three ; a truth Bedürfen, to want.

Lachen, to laugh. which is easy to demonstrate by analysis, in like manner. Let Begehren, to desiro.

Pflegen to foster. as examine a rectangle, whose sides are p and q; its surface, Brauchen, to use.

Schonen, to spare. which must be a function of p and q, we shall represent by $? Entbehren, to need.

Spotten, to mock. (P,9). If we oxamine another rectangle, whose dimensions are Entrathen, to do without. Verfehlen, to miss, or fail p+ p' and q, this rectangle is evidently composed of two others; Ermangelu, to want, or be with. Vergessen, to forget. of one having p and q for its dimensions, of another having på

out. Wahren, to guard. and q; so that we may put $:(p+p', 9) = 9: (p, q) + $:(P', 9). Erwähnen, to mention.

Wahrnchinen, to observe. Let p'=p; we shall have $ (2 p, q) = 20 (P,9). Let p' = 2 P; Gedenken, to think, or ponder. Walten, to manage. we shall have $ (3 p, q) = + (p, q) +$ (2 p. 9) = 30 (p, q). Let Genießen, to enjoy.

Warten, to attend to, or mind. p=3p; we shall have $ (4 p, q) = (p, q) + ° (3 P, 9) = 4¢ Gewahren, to observe. (p, q). Hence generally, if k is any whole number, we shall have • (k P, 9) = k $ (p, q) or $ (p, q) = P(k p, I); from which it fol


Bedürfen, begehren, brauchen, entbehren, erwähnen, genießen, pflegen lows that ° (p, q) is such a function of p as not to be changed by schonen, verfehlen, vergessen, wahrnehmen, waren and warten, take more

P substituting in place of p any multiple of it k p. Hence this harren and warten are more commonly construed with auf, and

frequently, in common conversation, the accusative. Achten, function is independent of p, and cannot include any thing except lachen, spotten and walten with über, before an accusative. q. But for the same reason

°(p, q) must be independent of g;

§ 126. RULE.
hence ° (p, q) includes neither p nor q, and must therefore be
P 9

The following reflexive verbs take, in addition to the pronoun limited to a constant quantity a. Hence we shall have (P, 9) peculiar to them, a word of limitation in the genitive: = a pq; and as there :s nothing to prevent us from taking a =?: Sich anmaßen, to claim. we shall have (P. 9) =Pq; thus the surface of a rectangle is

Sich erfrechen, to presuma. equal to the product of its two dimensions.

, annehmen, to engage in.

erinnern, to remember. In the very same manner, we could show, that the solidity of

bedienen, to use.

crfühnen, to venture. a right-angled parallelopipedon, whose dimensions are p, q, r, is

befleißen, to attend to.

" erwehren, to resist. equal to the product par of its three dimensions.

befleißigen, to apply to.

freuen, to rejoice. We may observe, in conclusion, that the doctrine of functions, begeben, to yield up.

getrösten, to hope for. which thus affords a very simple demonstration of the fundamen

. bemachtigen, to acquire.

rühmen, to boast. tal propositions of geometry, has already been employed with bemeistern, to seize. success in demonstrating the fundamental principles of Mechanics.

schamen, to be ashamed. See the Memoirs of Turin, vol. ii.

bescheiden, to acquiesce in. überheben, to be haughty. i besinnen, to ponder.

unterfangen, to undertake. entäußern, to abstain.

unterwinder, to undertake. entblöden, to dare, or be bold.


presume. entbreiten, to forbear.

versehen, to be aware. $ 124. RULE.

enthalten, to refrain.

webron, to resist. entschlagen, to get rid.

, weigern, to refuse. A NOUN limiting the application of an adjective, when in entsinnen, to recollect.

o wundern, to wonder. English the relation would be expressed by such words as of or erbarnien, to pity.

k P

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(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

to which an action is performed, will serve, also, to explain all versonals :

such examples as these: Ihnen bedeutet dieses Opfer nichts, to you Es gelüftet mich, I desire, or am pleased with. (i. e. so far as you are concerned) this sacrifice means nothing; Es jammert mich, I pity, or compassionate.

die Thränen, die Gurem Streit geflossen, the tears which have flowed Es reut mich, I repent, or regret.

in relation to (i. e. from) your dispute ; mir todtete cit! Souß das Es lohnt sich, It is worth while.

Pferd, a shot killed a horse for me, i. e. killed my horse ; falle

mir nicht, Kleiner, fall not for me, little one. In such instances as § 127. RULE.

the last two, the dative is often omitted in translating. The verbs following require after them a genitive denoting a

(4) The rule comprehends all such verbs as the following: thing and an accusative signifying a person :

antworten, to answer; danfen, to thank; bienen, to serve; drohen, to

threaten ; fehlen, to fall short; fluđen, to curse; folgen, to follow ; Anflagen, to accuse. Entwöhnen, to wean.

fröhnen, to do homage; gebütren, to be due ; gefallen, to please ; Velehren, to inform. Lossprechen, to acquit.

gehören, to pertain to; gehorchen, to obey; genügen, to satisfy; geBerauben, to rob. Mahnen, to remind.

reichen, to be adequate ; gleichen, to resemble; helfen, to help, &c. Beichuldigen, to accuse. leberführen, to convict.

(5) This rule, also, comprehends all reflexive verbs that Ontbinten, to liberate. Ueberheben, to exempt.

govern the dative: as, ich maße mir feinen Titel an, welchen ich nicht Gutblößen, to strip. Ueberzeugen, to convince.

habe, I claim to myself no title which I have not; as, also, Entheben, to exempt. Bersichern, to assure.

impersonals requiring the dative: as, e beliebt mir, it pleases me, Entladen, to disburden.

Vertrösten, to amuse, or put off or I am pleased ; es mangelt mir, it is wanting to me, or I am Entfleiden, to undress.

with hope. wanting, &c, Entlassen, to free from. Würdigen, to deem worthy.

(6) The dative is also often used after passive verbs: as, Gntledigen, to free from

Zeihen, to accuse; to charge. ihnen wurde wiederstanden, it was resisted to them, i. e, they were Entsegen, to displace.

resisted : von Geistern wird der Weg dazu bescüßt, the way thereto is

guarded by angels; ihm wird gelohnt, (literally) it is rewarded to Examples.

him, i. e. he is rewarded. Er hat mich meines Geldes beraubt, he has robbed me of my money. Der Bischof hat den Prediger seines Amtes entfeßt, the bishop has

$ 130. RULE. removed the preacher from his office.

Many compound verbs, particularly those compounded with OBSERVATIONS

er, ver, ent, an, ab, auf, bei, nac, vor, fu and wider, require

after them the dative; as, (1) The verbs above, when in the passive voice, take for their Ich habe ihm Geld angeboten, I have offered him money. nominative the word denoting the person, the genitive of the thing remaining the same; as, er ist eines Verbrechens angeflagt

$ 131. Rule. worden he has been accused of a crime.

An adjective used to limit the application of a noun, where in $ 128. RULE.

English the relation would be expressed by such words as

or for, governs the dative : as, Nouns denoting the time, place, manner, intent or cause of an action, are often put absolutely in the genitive and treated as

Sei teinem Herrn getreu, be faithful to your master. adverbs: as,

Das Wetter ist uns nicht günstig, the weather is not favourable Des Morgens gebe ich aus, in the morning I go out. Man sudit ihu aller Orten, they seek him everywhere.

OBSERVATIONS. Id bin Willens hinzugehen, I am willing to go there.

(1) Under this rule are embraced (among others) the followOBSERVATIONS.

ing adjectives : ähnlich, like ; angemessen, appropriate ; angenchm, (1) This adverbial use of the genitive is quite common in agreeable; anstößig, offensive; beannt

, known; beschieden, destined; German. In order, however, to express the particular point, or

eigen, peculiar; fremd, foreign ; gemäß, according to; gemein, comthe duration of time, the accusative is generally employed, or a lieb, agreeable; nahe, near; überlegen, superior ; willkommeu, wel.

mon; gewachsen, competent; gnädig, gracious; heilsam, healthful ; preposition with its proper case ; as, Ich werde nächsten Montag aus

come; widrig, adverse ; dienstbar, serviceable; gehorsam, obedient ; der Stadt gehen, I shall go out of town next Monday.

nüßlicy, useful. $ 129. RULE.

§ 132. Rule A noun or pronoun used to represent the object in reference

A noun or pronoun which is the immediate object of an ucija to which an action is done or directed, is put in the dative : as, transitive verb, is put in the accusative : Ich danke dir, I thank (or am thankful to) you.

Wir lieben unsere Freunde, we love our friends. Er gefällt vielen Leuten, he pleases many people.

Der Hund bewacht das Haus, the dog guards the bouse. Er ist dem Tode entgangen, he has escaped from death.


(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 nther respects, that have a transitive sig. sented as acting. Compared with the accusative, it is the case nification. Accordingly, under this rule come all those imperof the remole 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 all those verbs having a causative signification, as, fällen, to fell, einen Brief, I wrote (to) my father a letter, the immediate object i. e. to cause to fall; as also nearly all verbs compounded with is a letter ; while father, the person to whom I wrote, is the the prefix be. The exceptions are, begegnen, behagen, bestehen, be remote object. The number of verbs thus taking the accusative harren and bewachsen. with the dative, is large.

(2) Lehren, to teach ; nennen, to naine ; heißen, to call; schelter, to (2) On the principle explained in the preceding observation reproach (with vile names); taufen, to baptize (christen) ; take may be resolved such cases as the following: es tut mir leid, it after them two accusatives : as, er lehrt mich die deutsche Spracy, he causes me sorrow, or I am sorry, es wird mir im Herzen wehe thun, it teaches me the German language ; er nennt ihn seinen Kettet, he will cause pain to me in the heart (it will pain me to the heart). calls him his deliverer. See Sect. 53

to us.

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