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in the one case within, and in the other without, the triangle upon the base and the base extended.

(The triangles will be readily constructed; or, see Snowball's Trigonometry, page 70, figures 1, 2, and 3, reversing the positions of A and B.)

Let the three angles of each triangle be indicated by the letters A, B, C, and the sides opposite them by a, b, c. From the first triangle we have, a? 62 + c 2 A B. A D

(Euc. ii. 13) But A D = b X cos. A

.. a? 32 + ca 2 b c. cos. A also from the second,

a? - b2 + ca + 2 A B. A D (Euc. ii. 12) But A D = b cos. CAD=- - 6 X cos. CAB

6 x cos. A

= 12 + ca — 2 b c. cos. A, as before. From each of these expressions we have,

62 + c a? Cos. A =

2 b c
In Section iv. 1, it was shown that
Cos 2 A=1—2 sin. 2A

If for 2 A we put A, and for A, then


Cos. A = 1- 2 sin.?

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Substituting this value of cos. A,

А 62 + c a? 1- 2 sin.


2 b c A

12 + c - a?
.. 2 sin.?

a 72 ca + b c a - (6c)2
2 b c

2 b c
(a + bc) (

ab+c) 2 b c

2 b c

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A (a + bc)(a - b + c) .. sin.?

4 b c A ✓ (a + bc)(a b + c) ... sin. 2

4 b c This expression is further simplified by putting S= a + b + c

= half the sum of the sides. 2 And 2 S= a + b + c ..a + bc=2(S-c) and a-b+c=2(S-1)

A ✓ (S — 6) (S -c) ... sine

(1) 2 In a similar manner it is shown that

= +

(2) 2

be Multiply (1) by twice (2)

A A 2
2 cos.


✓ S(S—a)(S—6) (S—c) 2.

A A But sin. A = 2 sin.



2 .. sin. A = ✓ S(S—a)(S—b)(s—c)....(3)

be Again, area A BC=AB X CD=1cX 6 X sin. A For sin. A substitute its value (3)


2 .. area = cox ve v S(S—a) (S—b) (S—c)

✓S (S-a) (5-6) (S(6( The logarithm of a number to any assumed base, is the quantity expressing the power to which that base must be raised to produce the given number. The base most generally adopted is 10, so that the logarithms of numbers given in our tables are the powers to which 10 must be raised to produce these numbers.


Thus the logarithm of 100 is 2, as 10 = 100; the log. of 1000 3, since 103

1000 ; the log. of numbers between 100 and 1000 is 2 plus a fraction; the log. of numbers between 10 and 100 is 1 plus a fraction; the log. of numbers between 10 and 1 is some fractional quantity less than 1. The properties of logarithms of greatest practical utility in calculation are—1st. The logarithm of a product is equal to the sum of the logarithms of its factors; so that to multiply numbers we have only to add their logarithms and the number corresponding to the sum of their logarithms will be the product of the numbers. 2nd. To divide one number by another we have only to deduct the log. of the divisor from that of the dividend.

3rd. The logarithm of any power of a number is equal to the product of the logarithm of the number by the index of the power.—4th. The logarithm of the root of any quantity is equal to the logarithm of the quantity divided by the index of the root.

Hence, if we have to multiply one number by another, or to divide one by the other, we have only to add or subtract their logarithms : in the tables corresponding to the sum or difference of logarithms thus obtained are the product or quotient required. Or if we wish to get any power or root of a number we mu)tiply or divide its logarithm by the index of the given power or root and the number corresponding to the logarithms thus resulting are the power or root required. So that addition of logarithms is multiplication in common arithmetic; subtraction is division ; multiplication is involution ; and division is the extraction of roots.


1. “Prove the expression for the circumference and area of a circle in terms of its radius."


See Playfair's Euclid, Supplement, Book i. Propositions 5 and 9; also Tate's Geometry, Theorems 72, 76.

2. “Show that the solid content of any cone, or pyramid is found by multiplying the area of its base by one third of its height.”

See Playfair's Euclid, Sup. Book iii. 15 and 18. Tate's Geometry, Theorems, 91, 92.




1. Define the unit of work, and show that if a pressure

of m pounds be exerted over a space of n feet, the number of units of work done is represented by

m X n. 2. A locomotive engine working at 40 H.P. ascends

an incline of 1 in 250 steadily at the rate of 25 miles per hour ; what is the weight of the train ?

Ith 3. The traction of a waggon upon a level road is

of the gross load W; there is an ascent on this
road of 1 in n; show that if the friction on the
ascent be supposed the same as that on the level,
the traction P up it is represented by the formula

P= W +



1. There is a fall upon a stream of 11 feet, down which

22400 lbs. of water descend per minute, and on which there is erected a water-wheel whose modu

lus is :6; what is its horse power ? 2. A well 100 feet deep and 5 feet in diameter is to be

deepened 30 feet. Two men are employed at the bottom. The material is such that four times as much time is employed in the use of the pick as

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