

MATHEMATICS. 



ment, (which we suppose to be a given 

 point on a given line) measure the same 

 space on the segment. The two points 

 thus ascertained on the segment will 

 shew an angle of 27 degrees ; which will 

 be better seen by drawing lines from 

 them respectively to the centre where 

 the segment was described. When the 

 angle is to be more than 60 degrees, 

 another operation on a second line, made 

 at 60 degrees, will give the angle requir- 

 ed: thus you may make an angle of 60 

 degrees in the intended direction ; and if 

 the whole angle to be made amount to 

 73, you may add a second angle of 13. 

 But the neatest and shortest way is to 

 draw a perpendicular to the given line, 

 on the point whence the segment arises, 

 and from that to make an angle equal to 

 the complement : thus, if the angle is to 

 be 73, from the base line, you should 

 make an angle equal to 17, which added 

 to 73 complete 90 degrees, and thus ob- 

 tain the desired angle by inversion. 



" A line being given, to find the sine 

 of a segment whose radius shall be the 

 hypothenuse of a triangle (at any given 

 angle), formed by that line, as a base, and 

 by the sine as a perpendicular thereto," 

 (fig. 3.) Here we have one of the most 

 important, yet simple, operations in ma- 

 thematics ; viz. the ascertaining a sine 

 upon an undescribed segment. Let the 

 base line, A B, be 174, and the given 

 angle be 42; make the angle at one end, 

 B, of the base, and at the other, A, raise 

 a perpendicular which is to become the 

 sine, when intercepted by the hypothe- 

 nuse C B. Take 174 from the line of 

 equal parts on your compasses, and open 

 your sector until the distance between 

 48 and 48 on the lines of sines corres- 

 ponds therewith. Now measure the dis- 

 tance between 42 and 42 on the lines of 

 sines, and their result, 162, will be the 

 length of the sine to a segment, of which 

 the hypothenuse of the triangle is radius, 

 and whose versed sine will be found by 

 continuing- the base line until it meets the 

 segment : the base line in this case will 

 be equal to the co-sine ; since a perpen- 

 dicular raised at the angular point paral- 

 lel to the sine, A C, would, if the seg- 

 ment were continued thereto, complete 

 the quadrant of a circle. 



But if, instead of taking the hypothe- 

 nuse for a radius, we take only the length 

 of the base line ; and from the same 

 point as before, draw a segment, A D, 

 from the end of the base to the hypothe- 

 nuse ; then, Instead of being a sine, the 

 line whose length we have just ascertain- 



ed to be 162 will be a tangent, and comes 

 under the next example. 



" To ascertain the length of a tangent 

 under a given angle, on a given line.'* 

 Take the distance 174 (equal to the ra- 

 dius), from the line of equal parts, and 

 open your sector, so that it may be the 

 distance between 45 and 45 on the lines 

 of tangents. Then take the distance 

 from 42 to 42 on the same lines, and it 

 will be found equal to 162 on the line of 

 equal parts. Hence w r e see ttiat the tan- 

 gent of a segment made on the base as a 

 radius is the line of a segment made on 

 the hypothenuse as a radius ; the angle 

 in both instances being the same, and not 

 exceeding 45. 



" To find the length of the secant in 

 the same figure." Take the length of 

 the base, as before, from the line of equal 

 parts, and spread the sector until that 

 measure reaches from to (that is, from 

 the very beginning) of the lines of se- 

 cants ; measure the distance from 42 

 to 42 on the lines of secants; it will 

 reach to 238 on the line of equal parts, 

 and give that for the length of the hypo- 

 thenuse, which is in this case considered 

 as a secant. 



Besides the lines already described, 

 there are some that require the sector to 

 be completely unfolded, so as to be all 

 in one line. These are the artificial lines 

 of numbers, sines, and tangents, taken 

 from Gunter's tables, which depend on 

 logarithms for the solution of their opera- 

 tions ; as will be seen under the head of 

 NAVIGATION, in which the properties of 

 Gunter's scale are illustrated. 



MATHEMATICS, originally signified 

 any discipline or learning ; but, at pre- 

 sent, denotes that science which teaches, 

 or contemplates, whatever is capable of 

 being numbered or measured, in so far 

 as it is computable or measurable ; and, 

 accordingly, is subdivided into^arithmetic, 

 which has numbers for its object, and 

 geometry, which treats of magnitude; 

 See ARITHMETIC and GEOMETRY. 



Mathematics are commonly distinguish' 

 edinto pure and speculative, which con- 

 sider quantity abstractedly ; and mixed, 

 which treats of magnitude as subsisting in 

 material bodies, and consequently are in- 

 terwoven every where with physical con- 

 siderations. Mixed mathematics are very 

 comprehensive ; since to them may be re 

 ferred astronomy, optics, geography, hy- 

 drography, hydrostatics, mechanics, forti- 

 fication, navigation, Sec. See ASTRONOMY 

 OPTICS, &c. 



Pure mathematics have one peculiar 



