The division of the circumference of a circle, which we have noticed, into 360 degrees, &c. is called the sexagesimal division, and was that adopted by the ancients. Soon after the restoration of the sciences in Europe, many eminent mathematicians recommended the adoption of the centesimal division, by which the circumference is divided into 100 degrees, each degree into 100 minutes, &c. Our countryman Briggs, to whom mathematicians have been so much indebted for his laborious calculations of trigonometrical and logarithmic tables, afterwards attempted to get this division generally adopted; and had his attempt been somewhat earlier, it might probably have succeeded; but just at that period some trigonometrical tables were published by Vlacq, of greater extent than had been previously attempted, in which he had adopted the sexagesimal division; and at a time when mathematical tables were comparatively scarce, and the methods of calculating them extremely laborious, it was probably felt as too great a sacrifice to the advantage of the centesimal division to render these tables comparatively useless by its adoption, in addition to those inconveniences respecting the works of previous authors, which must necessarily be the consequence of a change from the one system to the other. From whatever cause, however, the attempt to introduce the new division of the circle failed, as did also another plan, which was suggested about the same time for preserving the sexagesimal division of the circle into 360 degrees, and introducing the centesimal division of each degree into 100 minutes, each minute into 100 seconds, &c. Another attempt was made in France at the time of the revolution to introduce the centesimal division, which was adopted by many eminent mathematicians in that country. Their example, however, seems to have been little followed in other countries, but as a considerable number of scientific works have been published in the French language in which this division of the circle is made use of, we shall give the rules for converting degrees, &c. in one scale to those of the other, with a few examples which will best explain the advantages of the centesimal division in the abbreviation of arithmetical computations. (11.) Let F represent the number of degrees contained in a given angle in the centesimal division: E the number according to the sexagesimal division. Ex. Convert 7° 15' 2" of the centesimal scale into degrees, minutes, &c., of the sexagesimal scale. Ex. Find the number of degrees, &c., in the centesimal scale corresponding to 3° 5' 33" of the sexagesimal scale. We must first convert 5' 33" into the decimal of a degree as follows: These examples show obviously the advantage of the centesimal division in the superior facility it affords of expressing minutes, seconds, &c., in the decimal notation, which is done merely by inspection, without the labour of successive division, as in the sexagesimal scale. SECTION II. Definitions of certain functions of an Angle, and of the Arc subtending it-Fundamental formula-Conventional use of the negative sign--its utility in Trigonometry. (13.) Let C L be a fixed line passing through the given point C; and let the line C K form the angle LCK (which we will denote by the letter A) with C L. With any radius CA describe an arc of a circle about C as centre, meeting C L in A and CK in P, and draw PM perpendicular to CA. The ratio of PM to CP is called the sine of the angle A; and the ratio of CM to CP is called the cosine of that angle; or as they are usually written The ratio which the sine of an angle bears to its cosine is called the tangent of the angle. The inverse of this ratio is called the cotangent. The ratio of unity to the cosine of an angle is denominated the secant; and that of unity to the sine, the cosecant. The difference between unity and the cosine is called the versed-sine. The difference between unity and the sine of an angle is called the coversed-sine. (15.) Thus each of the trigonometrical functions of an angle above defined is easily expressed in terms of the sine of that angle; and in a similar manner each might be expressed in terms of any other. *In some treatises, (sin A) is denoted by sin "A. This latter notation is however now fallen in some degree into disuse, it being used to denote an operation of a different kind. (16.) The sine, cosine, tangent, &c., as above defined, are functions of the angle, and are quite independent of the absolute length of the arc subtending it, or of the radius of that arc. The following are definitions of functions of the arc: Having a figure ACP as above, complete the quadrant APB, draw AT perpendicular to the radius C A from its extremity A, meeting the line CP produced indefinitely in T, and B t, Pm perpendiculars to the rad. C B, meeting the same line in t. Draw MP perpendicular to AC and join P, A. MP is called the sine of the arc AP B M: (17.) If we denote the length of the arc A P subtending the angle A, by a, and that of the radius CP by r, we have |