INTRODUCTION. LEMMA Ι. 1. AN angle at the centre of a circle is to four right angles, as the arc on which it stands is to the whole circumference of the circle. Let ABC be an angle at the centre of the circle ACF, standing on the arc AC; the angle ABC: four right angles :: arc AC: whole circumference ACF. D C H Produce AB till it meet the cir- E cumference in E, and through B draw DBF perpendicular to AE. Because ABC, ABD are two angles at the centre of the circle ACF, the angle ABC: angle ABD :: arc AC: arc AD (33. 6);* therefore the angle ABC: four times the angle ABD :: arc AC: four times the arc AD (4.5). But ABD is a right angle, therefore four times the arc AD is equal to the whole circumference ACF. Hence the angle ABC: four right angles :: arc AC: whole circumference ACF. 2. Cor. 1. Equal angles at the centres of different circles stand on arcs which have the same ratio to their circumferences. If the angle ABC at the centre of the circles ACF, GHK, stand on the arcs AC, GH, then the arc AC: whole circumference of the circle ACF :: angle ABC: four right angles; and the arc GH: whole circumference of the circle GHK :: angle ABC: four right angles. Therefore, &c. * The references to the elements of geometry in this work are made to Playfair's Geometry, which is a new edition of Euclid's Elements, with great additions and improvements. A 3. COR. 2. Hence the arcs which subtend equal angles at the centres of different circles are to one another as the circumferences of the circles. The arc AC: circumference ACF :: arc GH: circumference GHK (11.5). 4. Cor. 3. Hence, if the circumferences of any two circles be divided into the same number of equal parts, whatever number of those parts is contained in any arc AC of one circle, subtending a given angle ABC, the same number of parts will be contained in the arc GH of the other circle, subtending the same angle ABC. 5. Cor. 4. Hence, if a circle of any radius whatever be divided into 360 equal parts, called degrees, each degree into 60 equal parts, called minutes; each minute into 60 equal parts, called seconds, &c., the number of degrees, minutes, and seconds, intercepted by two radii, BA, BC, will be a proper measure of the angle ABC. For the number of degrees, minutes, &c. so intercepted, will be to 360 degrees, as the angle ABC is to four right angles; therefore the number of degrees, minutes, &c. so intercepted, will be to the number of degrees, minutes, &c. intercepted by any other two radii, as the magnitude of the former angle to the magnitude of the latter. And this is the proper meaning of a measure. One quantity is said to be a measure of another, when the measure in one case is to the measure in any other, as the magnitude of the quantity to be measured in the former case is to the magnitude of the quantity to be measured in the latter case. 6. Since then the whole circumference of a circle consists of 360 degrees, a right angle will be 90 degrees, two right angles will be 180 degrees, half a right angle will be 45 degrees, each of the angles of an equilateral triangle will be 60 degrees. The magnitude or quantity of any arc or angle is denoted thus, 21d 16m 45s, or thus, 21° 16′ 45′′, and is read 21 degrees, 16 minutes, 45 seconds. Note. Some mathematicians advise to divide a degree into centesimal parts rather than sexagesimal; and it would perhaps be more convenient to divide not only a degree, but even the whole circumference of the circle, in a decuple ratio; which division has been adopted in France. The advantage which would accrue from the division of the circle in a decuple proportion arises chiefly from the nature of the notation now in use, by which the value of any number is augmented in a decuple proportion, and diminished in a subdecuple proportion, to which therefore such a division would be peculiarly adapted. The ancients chose the number 360, probably because the Egyptians, to whom this division of the circle is usually ascribed, thought that the year consisted of 360 days, or that the circle which the sun seemed to describe in a year was already distinguished by nature into 360 parts. Another convenience in this division is, that the number 360 is divisible by several numbers without a remainder, as by 10, 6, 4,2, 90, 60, 9, &c., by which means they could accurately express several parts of the periphery of the circle in whole numbers, which they considered far more clear, and less intricate than fractions. LEMMA ΙΙ. 7. ALL angles are to one another as their arcs directly; and their radii inversely. Fig. to Art. 1. The angle ABC: 4 right angles :: the arc AC: circumference ADEFA (1.), therefore the angle ABC= AC X 4 right angles. But four right angles are a constant circum. ADEFA quantity, and the circumferences of circles are as their diameters or radii (8. 1. Sup.). Consequently the angle ABC varies as AC AB Let ac be any other arc, abc any other angle, ab the radius; then, in the same manner, the angle abc will vary as Hence the angle ABC : angle abc :: AC ac : ac ab 1 TRIGONOMETRY. 8. TRIGONOMETRY is a branch of the general science of geometry, which treats of the properties and relations of certain straight lines drawn in and about a circle, and also teaches to compute the sides and angles of triangles by means of a set of tables called a Trigonometrical Canon. It is divided into two parts, Plane Trigonometry and Spherical. The former has for its object rectilinear triangles, the latter, triangles formed by the intersection of three great circles on the surface of a sphere. 9. In a practical sense, Trigonometry may be defined to be, the application of number to express the relations of the sides and angles of triangles to one another. It therefore supposes that the learner can perform the common numerical operations; and it borrows from arithmetic certain signs or characters which peculiarly belong to that science. Thus, the product of two numbers represented by A and B, is denoted either by A. B, or AxB; and the product of two or more numbers multiplied by one or more numbers, as A+B into C, or A+B into C+D, is expressed by (A+B) C, and (A+B) (C+D), or sometimes by A+B x C, and A+B x C+D. The quotient of any number A divided by any number B, A is expressed by B, or A÷B. The sign signifies the square root. Thus, M denotes the square root of M, or is a number which, if multiplied by itself, will produce the number M. Also, M2+N2 denotes the square root of M2 + N2. PLANE TRIGONOMETRY. 10. Plane trigonometry is the science which treats of the analogies of plane triangles, and of the methods of determining their sides and angles. It also comprehends whatever appertains to the properties and relations of certain straight lines drawn in and about a circle. 11. In this treatise Plane Trigonometry is divided into three sections. The first section explains the principles; the second contains the rules of calculation, the practical solution of the cases of rectilinear triangles, and the mensuration of heights and distances; the third contains the changes of the signs of trigonometrical lines, the investigation of some theorems necessary to the solution of the more difficult problems in Trigonometry, and the construction of trigonometrical tables. SECTION 1. ELEMENTS OF PLANE TRIGONOMETRY. DEFINITIONS. 12. About the centre C (Fig. 1), with the radius CA, describe a circle; produce AC till it meet the circle again in D, so that AD may be a diameter. Draw the diameter HL perpendicular to AC. The two diameters AD, HL will divide the circumference into four equal arcs, called quadrants, each arc containing 90 degrees (6). Draw tAT touching the circle in A; and draw the radius CB, and produce it till it meet the line tAT in T. 13. DEF. 1. The difference of any arc from a quadrant, or of any angle from 90 degrees, is called the complement of that arc or angle. Thus, let the arc AB be reckoned from A as its origin,* then HB is the complement of AB, and the angle HCB is the complement of ACB. 14. DEF. 2. The difference of any arc from a semicircle, or of any angle from 180 degrees, is called the supplement of that arc or angle. Thus, the arc DHB is the supplement of AB, and the angle DCB is the supplement of ACB. 15. DEF. 3. The chord of any arc is a right line drawn from one extremity of the arc to the other. Let the points B and E be in the circle, and join B, E; then the straight line BE is the chord of the arc BAE, or of the angle BCE, of which the arc BAE is the measure. 16. DEF. 4. The sine or right sine of any arc is a straight line drawn from the end of the arc perpendicular to the diameter passing through the beginning of the arc. From B demit BF perpendicular to AD, then BF is the sine of the arc AB, or of the angle ACB. * A geometrical circle has no beginning. But in the application of Trigonometry to most subjects, especially to astronomy, it is necessary to fix upon some point as a beginning, whence all arcs are to be computed. Thus, the degrees on the equator and the ecliptic are reckoned from one of the equinoctial points. |