Euclid's Elements of Geometry

PROBLEM II.

Of the three sides and three angles of any plain triangle, any three being given, whereof one at least is a side, to find the rest: or, all the angles being given, to find the ratios of the sides.

other

BC: AC: S. A: S. ABC (3. Pl. Tr). If the side BC angles opposite the given angle be and greater than the other given

side AC, the angle ABC is acute. But if BC be less than AC, since the sine of an angle and of its complement to two right angles is the same, by cor. 1. Def. PL Tr. ; the species of the angle B, namely, whether it be acute or obtuse, must be known, or the solution will be ambiguous. The angles A and ABC being given, the angle ACB may be found, being the complement of their sum to 180° or two right angles, by 32. 1 Eu.

Case.

Given.

Sought.

Solutions.

Two sides The other Let AC be the greater) and the in-angles. of the given sides, and cluded angle. The angles AC+B C: AC-BC : AC, BCA and ABC.T ABC+A: T ABC—A and the angle

ACB.

(6. Pl. Tr). Whence is found] the difference of the angles A and ABC, whose sum is given, being the complement of the given angle ACB to two right angles, whence the angles A and ABC may be found by prop. 7 of this.

Let a perpendicular CD be let fall from one of the angles ACB on the opposite side AB. And, by cor. 1. 5 and 6. 2 and 16. 6 Eu. in fig. 1 AB: AC÷CB::AC-CB: ADDB, and in the case of fig. 2. AB: ACCB:: AC-CB: AD BD, whence the sum and difference of AD and BD being in either case given, the lines AD, BD themselves may be found by prop. 7 of this; and thence, by case 2 of right angled triangles, the angles CAD, CBD, and of course, in fig. 2. the angle ABC, the complement of CBD to two right angles by 13. 1 Eu., may be found.

SUPPLEMENT

TO THE SIX FIRST BOOKS OF

EUCLID'S ELEMENTS OF GEOMETRY.

205

BOOK I.

ELEMENTS OF CONICK SECTIONS.

NOTE. In subsequent citations, sup. denotes supplement.

DEFINITIONS.

1. [See note.] Conick Sections, or Pattalloids, are figures formed by lines, the sums or differences of the distances of every point of which, from two given points, or the distances of every point of which from a given point, or from a given point and given right line, are equal. And these figures are sometimes, for brevity, called, sections.

Fig. 2. B C

Cor. 1. Right lines come within this definition. For the sums of the distances of every Fig. 1. C D point, as C and D (see fig. 1), in any right line AB, from its extremes A and B, are equal. And the differences of the distances of every point, as C and D (see fig. 2), in the production of any right line AB, from its extremes A and B, are equal.

Cor. 2. Circular lines come also within the definition. For the distances of every point thereof from a given point, namely, their centre, are equal (Def. 10. 1 Eu).

But, by the term, conick sections, are chiefly understood the figures, called Ellipses, Hyperbolas and Parabolas, which are defined in the 2d, 4th and 8th definitions following.

2. An Ellipse, is a conick section, bounded by a line (AMBN), the sums of the distances of every point of which, from two given points (E and F) within the same, are equal.

3. These two given points are called, the focuses; the point (C), wherein the right line (EF), joining the focuses, is bisected, the centre of the ellipse; any right line (QP), passing through the centre, and terminated both ways by the ellipse, a diameter; the points (Q and P), wherein it meets the ellipse, the vertices of the diameter; the diameter (AB) which passes through the focuses, the greater, transverse or principal axis, and its vertices (A and B), the principal vertices; that diameter (MN), which is at right angles thereto, the less or second axis. The distance (CE or CF) of the centre from either focus, the eccentricity of the ellipse.

4. A Hyperbola. is a conick section, formed by a line (AQ or BP), the differences of the distances, of every point of which from two given points, (E and F) on different sides thereof, are equal; and if from the two points (A and B), in the right line (EF) joining these given points, the difference of whose distances from the same points is equal to the giv-Q en difference of distances, two such figures (AQ and BP) be formed, these figures are called, opposite hyperbolas.

5. The two given points (E and F) are called, the focuses; the point (C), wherein a right line joining them is bisected, the centre of the hyperbola or opposite hyperbolas; any right line (QP), passing through the centre, and terminated both ways by the opposite hyperbolas, a transverse diameter; the points (Q and P), wherein it meets the hyperbolas, the vertices of the diameter; the diameter (AB), which produced passes throughthe focuses, the transverse or principal axis, and its vertices (A and B), the principal vertices; the right line, which passes through the centre, at right angles to the transverse axis, and the distance of either extreme of which from either of the principal vertices, is equal to that of either focus from the centre, the second axis. The distance (CE or CF), of the centre from either focus, the eccentricity of the hyperbola or opposite hyperbolas.

6. Four hyperbolas (AQ, BP, MO, NU) are said to be conjugate; when the transverse axis of two of them, is the second axis of the other two, and the contrary.

7. Any right line (QU), passing through the centre, and terminated both ways, by hyperbolas conjugate to those which pass through the principal vertices, is called, a second diameter. 8. A Parabola, is a conick section, formed by a line (BPS), the distances of every point of which, from a given point (F), and a given right line (DK), are equal.

9. That given point (F), is called the focus, and that given right line (DK), the "directrix of the parabola; every right line D (as KPQ), perpendicular to the directrix, is called, a diameter; the point (P), wherein it meets the parabola, the vertex of that diameter; the diameter (DBF), which passses through the focus, the axis of the parabola; and its vertex, (B), the principal vertex.

10. A right line (RPX, see the 3 prec. fig.), which meets a conick section in any point (P), and, being produced both ways, falls wholly without the section, is said to be a tangent to the section, or to touch it in that point.

11. But if a right line, meeting a conick section, is on one side of its concourse with the section, within, and on the other, without the section, it is called, a secant.

12. A right line (ST), drawn from any point (S) of a conick section, meeting a diameter (PQ) of the section, and parallel

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Euclid's Elements of Geometry: The Six First Books. To which are Added ...