82 LECTURE X. 



that the ground rises or falls one foot, or one yard, in ten, when the sine of 

 the angle of its inclination to the horizon is one tenth of the radius. Angles 

 of different magnitudes are indeed proportional to the arcs, and not to the 

 sines, so that in this sense the sine is not a true measure of the comparative 

 magnitude of the angle ; but in making calculations, we are more frequently 

 obliged to employ the sine or cosine of an angle than the angle or arc 

 itself. It is, however, easy to pass from one of these elements to the others, 

 by means either of trigonometrical tables, or of the scales engraved on the 

 sector. 



The sines, tangents, and secants laid down on the sector, may be em- 

 ployed according to the properties of similar triangles, in the computation 

 of proportions. The same purpose is answered by Gunter's scale, by the 

 sliding rule, and by the logarithmic circles of Clairaut and of Nicholson,* 

 which are employed mechanically in the same manner as a table of loga- 

 rithms is used arithmetically, the proportion of any two numbers to each 

 other being determined by the distance of the corresponding divisions on 

 the scale ; so that if we wish to double or to halve a number, we have only 

 to find the distance from 1 to 2, and to lay it off from the given number 

 either way. (Plate VII. Fig. 93, 94.) 



The measurement of angles is at once applied to the estimation of dis- 

 tances in the dendrometer or engymeter ;t a part of the instrument forms a 

 base of known dimensions, and the angle at each extremity of this base 

 being measured with great accuracy, the distance of the object may be 

 inferred from an easy calculation, or from a table. The most complete 

 instruments of this kind have tw T o speculums for measuring the difference 

 of the angles at once, in the manner of Hadley's quadrant. Telescopic 

 scales or micrometers are also sometimes used for measuring angles sub- 

 tended by distant objects, of which the magnitude is known or may be 

 estimated, for example, by the height of a rank of soldiers, and inferring 

 at once the distance at which they stand. 



Arithmetical and even algebraical machines, of a much more complicated 

 nature, have been invented and constructed with great labour and ingenuity ; 

 but they are rather to be considered as mathematical toys, than as instru- 

 ments capable of any useful application.^ 



An angle, when once measured, can be verbally and numerically de- 

 scribed, by reference to the whole circle as a unit : but for the identification 

 of the measure of a right line, we have no natural unit of this kind, and it 

 is therefore necessary to establish some arbitrary standard with which any 

 given lengths and surfaces may be compared. It might be of advantage in 

 the communication between different countries to fix one single standard to 

 be employed throughout the world, but this does not appear to be practi- 



* Hist, et Mem. de Paris, 1727, H. 142. Nich. Journal, v. 40. Ph. Tr. 1753, 

 p. 96 ; 1787, p. 246. 



f Pitt's Dendrometer, Repertory of Arts, ii. 238. Fallen's Engymeter, Zach. 

 Monatliche Correspondenz, vi. 46. 



J Napier's Reckoning Rods, Leup. Th. Ar. t. 13. Robertson on GunCer's 

 Scale, Ph. Tr. 1753, p. 96. Nicholson's Logistic Circle and Scales, Ph. Tr. 1787, 

 p. 246 ; Herschel's Description of Babbage's Calculating Machine, Transactions of 

 the Cambridge Philosophical Society, iv. 425. 



