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a+b+c a+b-c a+c-b b+a-c
For this very elegant formula, we are indebted to Simon Lhuillier.
PROBLEM II. Given the three sides BC= a, AC=b, AB = c, to determine the position of the point I, the pole of the circle described about the triangle ABC.
Put the angle ACI = x, the arc AI = CI = BI
; in the triangles CAI, CBI, by
admitted formulas, we shall have
tuting in this equation the values of cos C and sin C expressed in terms of a, b, c, and making, for the sake of abridgment,
which determines the angle ACI. It may be observed, that by reason of the isosceles triangles ACI, ABI, BCI, we have ACI=1(C+A−B); we might also have ACI (B+C-A), and BAI=(A+B-C). Hence result these remarkable formulas :
To these may be added the formula which gives cot S, and admits of being put under the form,
PROBLEM III. On the surface of the sphere, to determine the line, in which are situated the vertices of all the triangles, having the same base, and the same surface.
Let ABC be one of the spherical triangles, whose common base is AB =C, and given surface A+B+CTS. Let IPK be an indefinite perpendicular drawn from the middle of AB, then IP being taken equal to a quadrant, P will be the pole of the arc AB, and the arc PCD drawn through the points P, C will be perpendicular to AB. Put ID=p, CD=q; the right-angled triangles ACD, BCD, in which we have AC=b, BC=a, AD= p+ c, BD = p · e, will give cos a = cos q cos (pc), cos b = cos qcos (p+c). But we found above A
substitute in this formula the values cos a + cos b = 2 cos q cos p cos c, cos c = 2 cos2 c, sin b sin C = sin c sin B = 2 sin c cos c sin B ; we shall have
Moreover, in the right-angled triangle BCD, we have sin a sin B =
This is the relation between P and q, which must determine the line whereon are situated, all the points C.
Having produced IP by a quantity PK = x, join KC, and put KC =y; in the triangle PKC, which has PC- } q, and the angle KPC-p, the side KC will be found by the formula cos KC = cos KPC sin PK sin PC + cos PK cos PC, or
in which, by substituting, in place of cos q cos p, its value cot S sinc
9- cosc, we shall have cos y = sin x cos o + sin q (cos x — sin x cotS sin c). Whence it appears that taking cos x sin x cot S sin co, or cot x = cotS sin c, we shall have cos y = sin x cos c, and thus the value of y will become constant.
Hence, if after drawing the arc IP at right angles to the middle of the base AB, the part PK beyond the pole is taken, so that cot PK = cotS sinc, the vertices of all the triangles having the same base c, and the same surface S, will be situated on the little circle described from the point K as a pole, at a distance KC, such that cos KC = sin PK cosc.
For this beautiful theorem we are indebted to Lexell. (See Vol. V. Part I. of the Nova Acta Petropolitana.)
On Proposition 3, Book VII.
THIS Proposition may be more rigidly demonstrated by referring it to the Preliminary Lemmas in the following manner. We shall first shew that the convex surface terminated by the edges AF, BG, and by the arcs A u B, Fx G, cannot be less than the rectangle F ABGF, the corresponding part of the inscribed prism's surface.
Let S be the surface in question; and, if pos sible, suppose the rectangle AB × AF=S+M, M being a positive quantity.
Produce AF, the altitude of the prism and the cylinder, to a distance AF', equal to n times AF, n being any whole number: if the prism and the cylinder are at the same time produced, A the convex surface S', included between the edges AF', BG', will evidently contain n times the surface F' S; so that we shall have S'n S; and because n× AF = AF', we shall have AB × AF = n S+nM=S'+ n M. Now n being any whole number taken at will, and M a given surface, n may be assumed, so that n M shall be greater than twice the 2 Au B M
segment A u B, since for this purpose it is enough to make n
hence in that case, the rectangle AB × AF', or the plane surface ABG'F' must be greater than the enveloping surface, composed of the convex surface S', and of two equal circular segments A u B, F' x' Gʻ. But, on the contrary, the latter surface is greater than the former, by the first Preliminary Lemma; hence in the first place, we cannot have SABGF.
We shall shew, in the second place, that the same convex surface S cannot be equal to that of the rectangle ABGF. For, if possible, sup
pose that by taking AE AB, the convex surface AMK could be equal to the rectangle AFKE: through any point M in the arc AME, draw the chords AM, ME, and erect MN perpendicular to the plane of the base. The three rectangles AMNF, MEKN, AEKF, having the same altitude, are to each other as their bases AM, ME, AE. Now we have AM + MEAE; hence the sum of the rectangles AMNF, MEKN is greater than the rectangle AFKE. The latter is, by hypothesis, equivalent to the convex surface AMK, composed of the two partial surfaces AN, MK. Hence the sum of the rectangles AMNF, MEKN is greater than the sum of the corresponding convex surfaces AN, MK. Hence one at least of the rectangles AMNF, MEKN must be greater than the corresponding convex surface; a consequence at variance with the first part demonstrated above. Hence, in the second place, the convex surface S cannot be equal to that of the corresponding rectangle ABGF.
It follows that we have SABGF, in other words, that the convex surface of the cylinder is greater than that of any inscribed prism. By the very same mode of reasoning, we could prove the convex surface of the cylinder to be less than that of any circumscribed prism.
On the Equality and the Similarity of Polyedrons.
At the beginning of Euclid's Twelfth Book, we find definitions 9 and 10 conceived in these terms:
9. Two solids are similar, when they are bounded by the same number of planes similar each to each.
10. Two solids are equal and similar, when they are bounded by the same number of planes equal and similar each to each.
As the object of these definitions is one of the most difficult points in the Elements of Geometry, we shall examine it somewhat in detail, discussing at the same time the observations which Robert Simson has offered with regard to it, in his edition of Euclid, p. 388 et seq.
In the first place, we may remark with Simson, that definition 10 is not properly a definition, but in truth a theorem which needs to be demonstrated; for it is not evident, that two solids are equal from the circumstance alone that their faces are equal; and if this proposition is really correct, it ought to have been proved either by superposition, or in some other way. It afterwards appears, that Def. 9 is infected with the error of Def. 10. For if Def. 10 is not demonstrated, it is possible to imagine that two unequal and dissimilar solids having equal faces might exist; and in that case, by Def. 9, a third solid having its faces similar to those of the first two might be similar to each of them, and would thus be similar to two bodies of different forms; a conclusion which implies a contradiction, or at least does not agree with the meaning naturally attached to the word similar.
Several Propositions of Books XI. and XII. in Euclid are founded on Definitions 9 and 10; among others, Prop. 28. XI., upon which depends the measurement of prisms and pyramids. It appears, therefore, that Euclid's Elements are liable to the objection of containing a considerable number of propositions which are not rigorously demonstrated. There is one circumstance, however, tending to weaken this objection, which must not be omitted.
The figures whose equality or similarity Euclid demonstrates, by means of Definitions 9 and 10, are such that their solid angles do not combine more than three plane angles; and, if two solid angles are composed of three plane angles equal each to each, it is proved with sufficient clearness, in various parts of Euclid, that those solid angles are equal. On the other hand, if two polyedrons have their faces equal or similar each to each, the homologous solid angles will be composed of the same number of plane angles equal each to each. So long, therefore, as no more than three solid plane angles are joined in each solid angle, the homologous solid angles will evidently be equal. But, if the homologous faces are equal, and also the homologous solid angles, there can be no longer any doubt that the solids are equal; for they will coincide if applied to each other, or at least be symmetrical. The enunciation of Def. 9 and 10 appears therefore to be true and admissible, at least in the case of triple solid angles, the only case which Euclid attends to, and hence the reproach of inaccuracy directed against this author or his commentators, loses much of its weight, applying only to the want of some explanations and restrictions which he has not given.
We have still to examine if the enunciation of Def. 10, which is true in the case of triple solid angles, is true in general. Robert Simson maintains that it is not, and that two unequal solid angles may be formed, both of which shall be bounded by the same number of faces equal each to each. In support of this assertion, he brings forward an example, which may be generalized as follows.
If to any polyedron we add a pyramid, giving it one of the polyedron's faces for a base; if afterwards in place of adding, we cut off such a pyramid, by forming in the polyedron a cavity equal to it, we shall thus obtain two new solids, both having their faces equal each to each, yet themselves unequal.
There can be no doubt as to the inequality of the solids so constructed; but we may observe, that one of those solids contains re-entrant angles; and it is more than probable that Euclid meant to exclude all irregular bodies which have cavities or re-entrant angles, limiting himself to convex polyedrons. Admitting such a restriction, without which, moreover, various other propositions would not be true, the example adduced by Simson, concludes nothing against the definition or theorem of Euclid.
But however this may be, it results from all those observations that Euclid's 10th and 9th Definitions cannot be retained in their present form. Robert Simson cancels the definition of equal solids, which ought certainly to have its place among the theorems; and he defines similar solids as those which, being included under the same number of similar planes, have their solid angles equal each to each. This definition is