pyramids coincide, the pyramids themselves will evidently coincide, and likewise the solid angles at their vertices. From this, some consequences are deduced. First. Two triangular spherical pyramids are to each other as their bases and since a polygonal pyramid may always be divided into a certain number of triangular ones, it follows that any two spherical pyramids are to each other, as the polygons which form their bases. Second. The solid angles at the vertices of these pyramids, are also as their bases; hence, for comparing any two solid angles, we have merely to place their vertices at the centres of two equal spheres, and the solid angles will be to each other as the spherical polygons intercepted between their planes or faces. The vertical angle of the tri-rectangular pyramid is formed by three planes at right angles to each other: this angle, which may be called a right solid angle, will serve as a very natural unit of measure for all other solid angles. And if So, the same number, that exhibits the area of a spherical polygon, will exhibit the measure of the corresponding solid angle. If the area of the polygon is 3, for example, in other words, if the polygon is of the tri-rectangular polygon, then the corresponding solid angle will also be of the right solid angle. PROPOSITION XXI. THEOREM. The surface of a spherical polygon is measured by the sum of all its angles, minus two right angles multiplied by the number of sides in the polygon less two. D From one of the vertices A, let diagonals AC, AD be drawn to all the other vertices; the polygon ABCDE will be divided into as many triangles minus two as it has sides. But the surface of E each triangle is measured by the sum of all its angles minus two right angles; and the sum of the angles in all the triangles is evidently the same as that of all the angles in the polygon; hence, the surface of the polygon is equal to the sum of all its angles diminished by twice as many right angles as it has sides minus two. 20 A B Scholium. Let s be the sum of all the angles in a spherical polygon, n the number of its sides; the right angle being taken for unity, the surface of the polygon will be measured by s-2(n-2), or s-2n+4. APPENDIX. THE REGULAR POLYEDRONS. A regular polyedron is one whose faces are all equal regular polygons, and whose solid angles are all equal to each other. There are five such polyedrons. First. If the faces are equilateral triangles, polyedrons may be formed of them, having solid angles contained by three of those triangles, by four, or by five: hence arise three regular bodies, the tetraedron, the octaedron, the icosaedron. No other can be formed with equilateral triangles; for six angles of such a triangle are equal to four right angles, and cannot form a solid angle (Book VI. Prop. XX.). Secondly. If the faces are squares, their angles may be arranged by threes: hence results the hexaedron or cube. Four angles of a square are equal to four right angles, and cannot form a solid angle. Thirdly. In fine, if the faces are regular pentagons, their angles likewise may be arranged by threes: the regular dodecaedron will result. We can proceed no farther: three angles of a regular hexagon are equal to four right angles; three of a heptagon are greater. Hence there can only be five regular polyedrons; three formed with equilateral triangles, one with squares, and one with pentagons. Construction of the Tetraedron. Let ABC be the equilateral triangle which is to form one face of the tetraedron. At the point O, the centre of this triangle, erect OS perpendicular to the A planc ABC; terminate this perpendicular in S, so that AS-AB; draw SB, SC: the pyramid S-ABC will be the tetraedron required. S B For, by reason of the equal distances OA, OB, OC, the oblique lines SA, SB, SC, are equally re S moved from the perpendicular SO, and B Construction of the Hexaedron. Let ABCD be a given square. On the base ABCD, construct a right prism whose altitude AE shall be equal to the side AB. The faces of this prism will evidently be equal squares; and its solid angles all equal, each being formed with three right angles: hence this prism is a regular hexaedron or cube. H D S C B A The following propositions can be easily proved. 1. Any regular polyedron may be divided into as many regular pyramids as the polyedron has faces; the common vertex of these pyramids will be the centre of the polyedron; and at the same time, that of the inscribed and of the circumscribed sphere. 2. The solidity of a regular polyedron is equal to its surface multiplied by a third part of the radius of the inscribed sphere. 3. Two regular polyedrons of the same name, are two similar solids, and their homologous dimensions are proportional; hence the radii of the inscribed or the circumscribed spheres are to each other as the sides of the polyedrons. 4. If a regular polyedron is inscribed in a sphere, the planes drawn from the centre, through the different edges, will divide the surface of the sphere into as many spherical polygons, all equal and similar, as the polyedron has faces. PLANE TRIGONOMETRY. In every triangle there are six parts: three sides and three angles. These parts are so related to each other, that if a certain number of them be known or given, the remaining ones can be determined. Plane Trigonometry explains the methods of finding, by calculation, the unknown parts of a rectilineal triangle, when a sufficient number of the six parts are given. When three of the six parts are known, and one of them is a side, the remaining parts can always be found. If the three angles were given, it is obvious that the problem would be indeterminate, since all similar triangles would satisfy the conditions. It has already been shown, in the problems annexed to Book III., how rectilineal triangles are constructed by means of three given parts. But these constructions, which are called graphic methods, though perfectly correct in theory, would give only a moderate approximation in practice, on account of the im perfection of the instruments required in constructing them. 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 trigonometrical lines, which furnish a very simple mode of expressing the relations 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. DIVISION OF THE CIRCUMFERENCE. I. For the purposes of trigonometrical calculation, the circumference of the circle is divided into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes; and each minute into 60 equal parts, called seconds. The semicircumference, or the measure of two right angles, contains 180 degrees; the quarter of the circumference, usually denominated the quadrant, and which measures the right angle, contains 90 degrees. II. Degrees, minutes, and seconds, are respectively desig : nated by the characters: ", '," thus the expression 16° 6' 15" represents an arc, or an angle, of 16 degrees, 6 minutes, and 15 seconds. III. The complement of an angle, or of an arc, is what remains after taking that angle or that arc from 90°. Thus the complement of 25° 40' is equal to 90°-25° 40′-64° 20′; and the complement of 12° 4′ 32′′ is equal to 90°-12° 4′ 32′′-77° 55′ 28′′. In general, A being any angle or any arc, 90°-A is the complement of that angle or arc. If any arc or angle be added to its complement, the sum will be 90°. Whence it is evident that if the angle or arc is greater than 90°, its complement will be negative. Thus, the complement of 160° 34' 10" is -70° 34' 10". In this case, the complement, taken positively, would be a quantity, which being subtracted from the given angle or arc, the remainder would be equal to 90°. The two acute 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 180°. Thus A being any angle or arc, 180°-A is its supplement. In any triangle, either angle is the supplement of the sum of the two others, since the three together make 180°. If any arc or angle be added to its supplement, the sum will be 180°. Hence if an arc or angle be greater than 180°, its supplement will be negative. Thus, the supplement of 200° is -20°. The supplement of any angle of a triangle, or indeed of the sum of either two angles, is always positive. GENERAL IDEAS RELATING TO THE TRIGONOMERICAL LINES V. The sine of an arc is ន the perpendicular let fall from one extremity of the arc, on the diameter which passes through the other extremity. Thus, MP is the sine of the arc AM, or of the angle ACM. The tangent of an arc is a line touching the arc at one extremity, and limited by the prolongation of the diameter which passes through the other extremity. Thus AT is the tangent of the arc AM, or of the angle ACM. B D R E P S A |