of the former parts is then measured upon the small, or sliding portion, and that length similar to the unit is divided into n equal parts. It is evident that, if the sliding-scale is moved so that the first lines of division in each will coincide, the distance moved over will be equal to the difference between the length of the re 1 1 1 (n-1) spective parts of division, or, - n n n n or ; &c. Hence, if A B be an inch of such a ale divided decimally and C D be nine-tenths of an inch upon the sliding scale also divided into tenths : then, if C D be moved along till the position of the figures 1 and 1 1 1 coincide, the scale will have slid ; if 2 and 102 100 2 3 2 coincide, then ; if 3 and 3, then 100 100 The advantage of the application of the vernier is obvious, where accuracy and minuteness of admeasurement are necessary, as in the case of the barometer, in which the division above is the one generally adopted. In its application to instruments for the measurement of angles the division usually adopted is that of 29 half degrees into 30 parts, when the coincidence of 1 2 the 1st and 2nd, &c. lines will indicate 60 60 1, 2, &c, minutes respectively. The same result is obtained when 19 third - degrees are divided into 20 equal parts. &c. or, 23 ALGEBRA. a X. Section I. 1. Multiply a a x + x2 by Multiply x +(a−b)x - a b by x + b. 2. What is the principle of subtracting a quantity con taining a minus term ? trate it simply. 3. Subtract a? (6—c*) from (a,b,c) and divide 23 -3 aể x—2 a' by x—2 a. SECTION II. 3 x2 + 6 x + 5 29 1. Shew that = 3 x - 6+ x + 4 x + 4 2 x and simplify 2 — 2 5 2. Prove the rule for dividing one fractional expression by another. Y 1 2 Divide ข Y ข• SECTION III. 1. Find the value of x in each of these equations 2 + = 2x - 7. 3 = 0. + X 2.C х a 2. What number is that, the double of which exceeds its half by 6? The difference of two numbers, and a quarter part of their sum are each equal to 2; find the num bers. 3. If A does a piece of work in 10 days, which A and B can do together in 7 days, how long would B take to do it alone ? Find the amount of P £ at compound interest for n years, the interest being paid yearly. a X. SECTION I. 1. (1) “Multiply a? - a x + 22 by (2) “Multiply xa + (a—b) x — ab by x + b." (1) ao a x + x2 a b 3 a 72 (2) x2 + (a - b) x — X + b a 62 x + a x 62 x 2. (1) “What is the principle of subtracting a quantity containing a minus term (2) “ Take 6 c from a, show the working and illustrate it simply.” (1) Consider the sign positive and proceed as in addition. (2) a (6-0) Here we have simply changed the signs of the quantities to be subtracted, and appended them to the a. Why the sign of b is changed is obvious; and why the minus sign of c is changed to plus is also easy of apprehension : for when b has been subtracted by changing its sign, too much by the amount c has been taken from a; because it was not b which had to be subtracted, but b less by c. Therefore c must be added to the result. Let a = 10; b = 8 and c = 5 10 — (8—5) 10 3 = = 7 10 —8 + 5 3. (1) “Subtract a? (62—c) from (a—5—c)?." (2) “ Divide 23 3 a' x 2 a3 by x 2 a." (1)(a-bmc)= {a —(} (a-6-c) =(a—) — 2 c(a—b) + c? a® + b2 + c 2 ab — 2 ac + 2bc a? — (62—c) = a? 22 + c 0 2 62 0 2 a b 2 ac + 2bc= 2 2 (62 — ab-ac+ bc) or = 7. { (مها) ه 3 x2 + 6 x + 5 1. (1) “Show that 29" = 3x - 6+ x + 4 +4 с (1) 3 x" (2) "Simplify 2 2 5 29 x + 4 24 29 * 6 x 3 x ll 3 X 2 (2) 2 (4 — 3 x) = 2 5 10 2. (1)“ Prove the rule for dividing one fractional quantity by another.” Y 1 1 (2) "Divide + Y x + y x + y” by X Y a. с a (1) Let 7 be a fraction which is to be divided by d We have to prove that, s === b Lets be divided by c, we have, another, à : х a a a b c But c was not the quantity by which we were to have divided, it was the dth part of c. Therefore the re |