The ratio of the cir to the radius ex 2π. Thus, the ratio of the whole circumference to the diameter, cumference though they are not commensurable, has been approximately determined by various methods, some of which will be conpressed by sidered hereafter. This ratio, which very frequently presents itself in analytical formulæ, is approximately expressed by the number 3.14159, and is also generally represented by the symbol T: the ratio of the circumference to the radius, which is the double of the ratio of the circumference to the diameter, is therefore expressed by 2*. The meaThe ratio of a quadrant to the radius, which is the measure sure of 90° is exof a right angle or 90°, is one-fourth part of the ratio of the pressed by circumference to the radius, and is therefore expressed by 2 Unity is the mea sure of an angle of 570.17'.45", and .0017453 is the mea sure of an ᅲ 2 If it be required to determine the angle whose measure is 1, or, in other words, the arc whose length is equal to the radius, it will be found to be 57°.2958 or 57°.17′.45′′ nearly: if the measure of an angle of one degree be required, it will be found П to be or .001745 nearly there exists therefore no simple 180 angle of 1o. numerical relation between the unit of measures and the unit of the angles which they measure. Angles and their measures are very often designated by the same symbols and the minations. 748. But though there exists no simple relation between angles and their measurest, by which they may be converted 1139 : 800 * Archimedes, in his Kúkλov Mérρnois, assigned as an approximate value of, which differs in excess from its true value by less thanth part of the diameter it is this value, which is generally used by workmen, in their estimation same deno- of circular work. Peter Metius, by a similar process, found the remarkable ratio 355 which is correct to 5 places of decimals, and which may be very easily remembered, by observing that if we write each of the three first odd digits twice in succession, as in the number 113355, the three last digits form its numerator and the three first its denominator. Later researches have assigned its value correctly to 208 places of decimals, a prodigious approximation, which is effected by processes which are extremely simple and expeditious, and well calculated to shew how much the most complicated calculations may be shortened by a judicious selection of formulæ. See Phil. Trans. 1841, p. 281. A right angle is capable of being determined geometrically, and is there. fore an invariable standard of angular magnitude, to which all other angles are referrible numerically, and some also geometrically: if we should call a right angle unity, and its centesimal subdivisions grades, minutes, seconds, we should have the French division of the quadrant, or of the right angle, with its scale of units increased one hundred-fold. There is no geometrical mode of determining an absolute unit of any other magnitude. Thus a linear unit must be sought for in some assumed standard, such promptly and easily into each other, yet in the case of certain periodical ratios, which will be considered in the next Chapter, which are equally determined by them, it is usual to represent them by common symbols, and to call them by common denomi 2 nations: thus, is considered equally as the representative of a right angle, and its measure: and similarly or (which are symbols very generally employed for such purposes) and any other symbols are applied indifferently to designate both angles or their measures. 749. Upon the same principle, if 0 is used to denote either Usual an angle or the measure of an angle, C B mode of denoting an angle and its comple ment. of the term 750. In speaking of geometrical angles and their comple- Extended ments, we assume them both to be less than 90°: denotes any goniometrical angle whatsoever, or its whether greater or less than 90°, we still continue to apply but if application measure, comple 0, though it may no longer lows, therefore, in conformity with the principles of Symbolical Algebra, that the meaning of the term complement, when no longer such as the English standard yard (which no longer exists), or by reference to some invariable standard in nature, such as the length of a pendulum vibrating seconds in vacuo in a given latitude at a given height above the sea, or to the length of a quadrant, or other definite part of the earth's meridian, such as the mètre of France: the same remark applies to the units of time, force, and all other physico-mathematical units. ment. Usual geometrical, is interpreted altogether with reference to the conditions which its symbolical representative is required to satisfy. 751. In a similar manner, if ◊ denote the angle BAP, or mode of de- its measure, then - is used to denote noting the supplement its supplement 180°-0 (PAb), or its meaof an angle. Various corres sure. In speaking of geometrical angles and their supplements, we suppose them both b B to be less than 180°: but T -0 continues to be called the supplement of 0, when is any goniometrical angle whatsoever. 752. The angle + 0, or its measure, corresponds to the same measures geometrical angle with the measure ± 2nπ ±0, where n is any ponding to whole number: in other words, if we merely regard the position geometrical of one line with respect to another, we may add to, or subtract the same angle. from, its measure, any multiple of 27, or of the measure of 4 right angles: this conclusion follows immediately from Art. 742. π In a similar manner, 0 corresponds to the same geome 2 responds to the same geometrical angle with ±2n«+w±0, or These equivalent measures, corresponding to the same geometrical angle, lead to the most important consequences, which will be more particularly considered in the subsequent Chapters. Great im753. The propositions in this and the following Chapter portance of the funda- will be found to constitute the grammar, as it were, of the lanmental propositions in guage of Trigonometry, and will be made the foundation of a Trigono- very extensive analytical theory, admitting of the most varied and important applications to every branch of mathematical and physical science. It is for this reason that the relations and signs of affection of goniometrical angles, and their measures, and of the periodical ratios (Chap. xxvI.) which determine them, minute metry. and unimportant as some of them may at first sight appear, will lead to consequences of very great interest and value*: they cannot therefore be too carefully studied and remembered. The whole theory of music is dependent upon the properties and conversions of a very small number of numerical ratios: yet how vast and complicated is the superstructure which is raised upon them! It should be the first lesson of a student, in every branch of science, not to form his own estimate of the importance of elementary views and propositions, which are very frequently repulsive or uninteresting, and such as cannot be thoroughly mastered and remembered without a great sacrifice of time and labour. CHAPTER XXVII. Ratios which determine, but do not measure, angles. Definition of the sine ON THE THEORY OF THE SINES AND COSINES OF ANGLES. 754. which is AN angle, such as BAP, is determined by its measure, (Art. 746): but there are other ratios which equally determine an angle, B If from any point whatsoever P, in one of the lines con and cosine taining the angle MAP or 0, we draw we shall form a right-angled triangle angle : then the ratio PM is called A M AM the sine, and the ratio is called the cosine of the angle AP PAM or 0: and inasmuch as these ratios remain unaltered whatever be the distance of the point P from A*, they are said to determine the angle, inasmuch as a definite value of the sine or cosine determines a definite value of the corresponding anglet. For if we take any other point whatsoever p in AP or in AP produced, and draw pm perpendicular to AM or to AM produced, we shall find, from a well-known pro P sine and cosine are always the same for the same angle: if therefore the angle be given, they will possess a determinate value. It was formerly the practice to define the sine and cosine as lines, and not as ratios: thus if we take an arc BP subtend ing an angle BAP (0) in a circle whose radius is M B and |