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giving the signs + and alternately to the sides, the first and the last will necessarily have the same sign, which makes one change fewer than the face has sides.
This being proved, let a represent the number of triangles, b the number of quadrilaterals, c the number of pentagons, &c, which compose the surface of the given polyedron; it follows, from what we have said above, that the total number of changes of sign to be found in going round each face, will not exceed 2 a for triangular, 4b for quadrilateral faces, 4 c for those of five sides, 6 d for those of six. Hence we shall have
N2a +46 + 4 c + 6d + 6e + 8f + 8 g + &c. Let A be the number of the polyedron's edges, H that of its faces; we shall have
2A 3a +46 +5c+6d+7e+8f+9g+ &c.
H= a+b+c+d+e+f+g+ &c.
But according to Euler's theorem, S + H = A + 2; hence 4 S= 8+ 4 A-4 H, and substituting,
4S 8+2a +46 + 6c+8d+ 10e + &c,
Compare this value to the limit found above; we find
But we cannot at once have N4 S, and N4S-8; hence it is impossible that the inclinations to the edges of the polyedron can vary all at once, without destroying the coherence of the planes which form the surface of the polyedron.
Suppose now that the inclinations to the edges do not all vary at once; that there are some which remain constant.
Let FI be one of these edges; we might imagine it to be suppressed, and the two adjacent faces FIG, EFIH combined into a single unplane face, terminated by the contour EFGIH whose form is invariable. Let us designate by A S', H' and A', what the numbers S, H, and Ai become after the suppression of an edge; we shall have H'H1, and A'A-1: also we have S'S, the number of solid angles being the same in both solids; hence we shall have S'+H'A'S+HA=2. Whence it is evident that Euler's theorem holds good likewise in the new solid, which contains one edge less, and one face less, two faces being combined into a single unplane face.
If from this second solid, there is cut off another of the edges on which the inclination remains invariable, the suppression of this edge will again occasion the reunion of two contiguous faces into one; and we shall be able to shew, as before, that Euler's theorem holds good in the third solid which results from the suppression of two edges.
We might continue to suppress as many edges as we please, provided this suppression carried with it the suppression of no solid angle; and Euler's theorem would always hold good in the remaining solid. This truth, indeed, may be seen directly and generally, from examining the
demonstration we have already given of Euler's theorem: which demonstration, in fact, does not suppose that the faces of the polyedron are plane; but would equally continue valid, though these faces were terminated by contours not situated in the same planes; it only supposes, that, as in our construction, each contour is represented by a spherical polygon, and that the sum of all the surfaces of these polygons is equal to the surface of the sphere. Nor is it even necessary that these polygons be all convex; it is enough that each of them may be regarded as the sum of several convex polygons; a condition which will always be observed, when, even by the suppression of several edges belonging to the given polyedron, several plane faces are combined into a single unplane face; because, in that case, the spherical polygon representing the latter face, will be composed of the sum of the convex spherical polygons which represented the suppressed plane faces.
Let us now proceed to the case where the suppression of the edges on which the inclination does not vary, carries with it the suppression of one or several solid angles, either because the inclinations to all the edges in each of these angles are invariable, or because these inclinations being capable of varying only on three edges, would, in that case, be of necessity invariable.
Suppose, first, that only one solid angle is suppressed; and let m be the number of faces in this angle, or the number of edges ending in its vertex. By suppressing the solid angle in question, we shall, at the same time, suppress m edges, and the m faces forming the solid angle will be reduced to a single face; hence, designating by S', A', H', what the numbers S, A, H become after the suppression of a solid angle, we shall have S' = S - 1, A = A' - m, H'H— (m - 1). From this we obtain S'+H'-A'S+H¬A=2; hence Euler's theorem still holds good in this new solid.
It is now evident, that in the given polyedron, as many solid angles as we please may be suppressed, and that Euler's theorem will always continue true with regard to the remaining polyedron; for by suppressing the solid angles one by one, we successively obtain different polyedrons, of which two consecutive polyedrons always fall under the case we have just examined.
Hence, generally, if in the proposed polyedron, all those edges are suppressed on which the inclination does not vary; whether by this suppression the number of solid angles remain the same, or become less, the remaining polyedron will always satisfy Euler's theorem; in other words, naming s, h, a, in this polyedron the quantities which correspond to S, H, A, in the proposed polyedron, we shall constantly have s+h―a=S+H¬A=2.
But in this last solid, the inclinations on the edges must all vary at once, all the edges on which the inclination does not vary being suppressed; hence this solid comes back under the first case; hence the simultaneous change of all the inclinations cannot take place without destroying the solid.
Hence, finally, no convex polyedron whatever can be changed into another convex polyedron, included under the same polygonal planes, arranged in the same order with regard to each other.
END OF THE NOTES.
TRIGONOMETRY has for its object the solution of triangles, that is, the determination of their sides and angles, when a sufficient number of those sides and angles is given.
In rectilineal triangles, it is sufficient to know three of the six parts which compose them, provided there be a side among these three. If the three angles only were given, it is obvious that all similar triangles would answer the question.
In spherical triangles, any three given parts, angles or sides, are always sufficient to determine the triangle; because, in triangles of this sort, the absolute magnitude of the sides is not considered, but only their relation to the quadrant, or the number of degrees which they contain.
In the Problems annexed to Book II., we have already seen how rectilineal triangles are constructed by means of three given parts. Propositions 24 and 25 of Book V. give likewise an idea of the constructions, by which the analogous cases of spherical triangles might be resolved. But those constructions, though perfectly correct in theory, would give only a moderate approximation in practice,* on account of the imperfection of the instruments required in constructing them: they are called graphic methods. Trigonometrical methods, on the contrary, being independent of all mechanical operations, give solutions with the utmost accuracy: they are founded upon the properties of lines called sines, cosines, tangents, &c. which furnish a very simple mode of expressing the relations that subsist between the sides and angles of triangles.
We shall first explain the properties of those lines, and the principal formulas derived from them; formulas which are of great use in all the branches of mathematics, and which even furnish means of improvement to algebraical analysis. We shall next apply those results to the solution of rectilineal triangles, and then to that of spherical triangles.
*We are naturally required to distinguish the figures which serve only to direct our reasoning in the demonstration of a theorem or the solution of a problem, from the figures which are constructed to find some of their dimensions. The first are always supposed to be exact; the second, if not exactly drawn, will give false results.
DIVISION OF THE CIRCUMFERENCE.
I. Till very recent times, geometers were agreed in dividing the circumference into 360 equal parts, called degrees, the degree into 60 minutes, the minute into 60 seconds, and so on. This method seemed to be of advantage in practice, on account of the great number of divisors which 60 and 360 have; but it was, in reality, subject to the inconvenience of complex numbers; and it frequently impeded the rapidity of calculation.
The mathematicians, to whom we are indebted for the new system of weights and measures, conceived that it would be of great advantage to introduce the decimal division in the measurement of angles. Upon this principle, considering as their fundamental unit the quarter of the circumference, or the quadrant, which measures the right angle, they have divided this unit into 100 equal parts, called degrees, the degree into 100 minutes, and the minute into 100 seconds.
We shall henceforth exclusively employ the new or decimal division of the circumference.* It is most agreeable to the nature of our arithmetic, and best fitted to abridge calcula
II. Degrees, minutes, and seconds, are respectively designated by these characters: °,', ": thus the expression 16° 6′ 75′′ represents an arc, or an angle, of 16 degrees 6 minutes 75 seconds. The relation of this same arc to the quadrant, taken for unit, would be expressed by 0. 160675. It likewise ap
*It is much to be regretted, that this centesimal division of the quadrant had not been proposed at an earlier period in the history of science. The advantages it has over the common or nonagesimal system are numerous and palpable; but as all our trigonometrical calculations have for so many centuries been performed, and their numerous results expressed, with a constant reference to the latter, the nonagesimal system has at length become as it were a part of the vernacular language of mathematicians; and bids fair to be as indestructible as a vernacular language. At all events, the French are as yet solitary in their rejection of it; and the general adoption of their new system, if it ever take place, which appears extremely problematical, must certainly be a very distant event. For these reasons, we could have wished that the nonagesimal scale had been followed in the present treatise; or that, without altering any material part of the work, we could have substituted it in our translation. This was found to be impracticable; but the present arrangement, while it exhibits the advantages of the new system, followed in all the most distinguished mathematical works of a recent date, will in other respects occasion very little inconvenience. The reader has only to bear in mind, that 200°, 100°, 50°, &c., when mentioned in the text, mean respectively 180°, 90°, 45°, &c., in the common tables; and that, generally, any angle may be reduced from degrees in the centesimal scale to degrees in the nonagesimal, by diminishing it in the ratio of 400 to 360, or of 10 to 9. Wherever this process could be attended with the slightest difficulty, or the omission of it might create any ambiguity, we shall subjoin a note, in which the required reduction will be exhibited.-ED.
pears, that the angle, measured by this arc, is to the right angle as 160675 to 1000000, a ratio which could not have been so easily deduced from the expressions furnished by the old division of the circumference.
Arcs and angles are, in calculation, expressed indifferently by numbers of degress, minutes, and seconds. Thus we shall designate the right angle, or the quadrant, by 100°; two right angles, or the semi-circumference, by 200°; four right angles, or the whole circumference, by 400°; and so on.
III. The complement of an angle, or of an arc, is what remains after taking that angle or that arc from 100°. Thus an angle of 25° 40', has for its complement 74° 60'; an angle of 12° 4' 62", has for its complement 87° 95′ 38′′.
In general, A being any angle or any arc, 100°-A is the complement of that angle or arc. Whence it is evident that, if the angle or arc is greater than 100°, its complement will be negative. Thus the complement of 160° 84' 10" is 60° 84/ 10". In this case, the complement, positively taken, would be the quantity requiring to be subtracted from the given angle or arc, that the remainder might be equal to 100°.
The two angles of a right-angled triangle, are, together, equal to a right-angle: they are, therefore, complements of each other.
IV. The supplement of an angle, or of an arc, is what remains after taking that angle or arc from 200°, the value of two right angles, or of a semi-circumference. Thus A being any angle or arc, 200° A is its supplement.
In any triangle, an angle is the supplement of the sum of the two others, since the three together make 200°.
The angles of triangles rectilineal and spherical, and the sides of the latter, have their supplements always positive; for they are always less than 200°.
GENERAL IDEAS RELATING TO SINES, COSINES, TANGENTS, &c.