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9. The angles of a triangle are in arithmetical progression, and the greatest of them is of the least. Find them.

10. The number of degrees in an angle is to the number of grades in its complement as 3 is to 5. Find the angle.

11. A, B, C are the three angles of a triangle. The number of degrees in A is equal to the number of grades in B and to the product of the number of degrees and of grades in C. Find A, B, C.

12. The side AB of a triangle ABC is produced to D, and AE is drawn making an angle with AC equal to BCA which is less than BAC. The angle CBD is 112° 11' 44"; and the angle EAB is 67° 48' 16". Find the angles of the triangle ABC.

13. What trigonometrical angles has the minute hand of a clock described since midnight at 3.15 a.m., at 7.40 p.m., and at 9.10 a.m., respectively?

14. ABCD is a square, AC and BD are its diagonals. Find the general expressions for the trigonometrical angles bounded by the lines

(1) AB and AC, (2) AB produced and BD. 15. Find in degrees the angles of

(1) an equilateral triangle, (2) a regular hexagon, (3) a regular pentagon, (4) a regular dodecagon, (5) a regular quindecagon.

16. Find the hypothenuse of the right-angled triangles whose sides are respectively, (1) 3 ft. and 4 ft. (2) 2 ft. 9 in. and 4 ft. 8 in.

(3) 4 ft. and 4 ft. 7 in.

17. Find the third side of the right-angled triangles whose hypothenuse and other side are, respectively,

(1) 26 in. and 10 in. (2) 4 ft. 5 in. and 2 ft. 4 in. (3) 7 ft. 1 in, and 3 ft.

The sides of a right-angled triangle are a62 and 2ab respectively. Find the hypothenuse.

18.

19. OA is a line of any length. AB is perpendicular to 0A, BC to OB, CD to OC, DE to OD &c., and

AB = BC = CD = DE = &c. = 0A. Find the lengths of the lines OB, OC, OD, OE &c. in terms of OA.

20. Show in the method of Euclid, Book I., that

(1) Two triangles, which are equal in area and have a side and adjacent angle of one equal to a side and adjacent angle of the other, are equal in all respects.

(2) Two triangles, which are equal in area and have two angles of the one equal to two angles of the other, each to each, are equal in all respects.

(3) Two triangles, which are equal in area and have a side and opposite angle of one equal to a side and opposite angle of the other, are equal in all respects.

(4) If two triangles, which are equal in area, have two sides of the one equal to two sides of the other, each to each, then the angle contained by the two sides of the one is equal or supplementary to the angle contained by the two sides of the other.

J. T.

2

CHAPTER II.

THE ANGLE AND THE CIRCLE.

§ 1. ON MEASUREMENT. 31. Two quantities are said to be homogeneous or comparable, when one is either equal to, less than, or greater than the other.

Thus, an inch and a mile—a degree and a grade-a triangle and a square—are pairs of quantities which are comparable in magnitude. But a line and an angle—a line and an area—an angle and an areaare pairs of quantities incomparable in magnitude.

Two homogeneous quantities are said to be commensurate, when the multiplication of the smaller by some finite number gives a quantity greater than the greater.

Thus, an inch and a mile are commensurate; but the space occupied by an atom and that occupied by the Solar System are practically incommensurate.

Two homogeneous commensurate quantities are said to be commensurable, when both are exact multiples of some common quantity commensurate with each.

Thus, 31 inches and 17} inches are commensurable; for 31 inches = 21 sixths-of-an-inch and '17} inches = 104 sixths-of-an-inch. Also 3/2 inches and 5/2 inches are commensurable. But 1 inch and 12 inches are incommensurable : for, since there is no fraction with integral numerator and denominator exactly equal to 1/2, 1 inch cannot be divided into any finite number of parts such that a number of those parts shall exactly equal 2 inches*.

* The distinction between commensurable and incommensurable is purely theoretical. Practically we cannot compare the measures of two quantities exactly; hence we can never determine whether they are really commensurable or not.

32. The definitions of the last article are equivalent to the following:

Two quantities are comparable, when the one has some ratio to the other: they are commensurate, when the ratio of either to the other is finite : they are commensurable, when their ratio is measured by a fraction with finite integral numerator and denominator.

33. A quantity of any kind is measured by assigning the ratio it bears to some known quantity of the same kind called the unit.

Thus we measure lengths by assigning their ratio to (say) a foot ; we measure angles by assigning their ratio to (say) a right angle.

The ratio of the given quantity to the unit is called the abstract or numerical measure of the quantity; or sometimes, more shortly, the measure of the quantity.

34. An accurate measurement of any quantity is practically impossible.

Suppose that we know that a magnitude lies between m 1 units and m+ 1 units. Then the error involved in taking m units for the true value is less than 1 unit. We should, therefore, say that to within 1 unit its measure is m units.

If another magnitude has to be found by multiplying the above by any number k, this second magnitude lies between km - k units and km + k units. The error involved in taking km units for the true value is less than k units. We should, therefore, say that to within k units its measure is kin units.

Therefore, the possible error in assigning the measure of a magnitude by multiplying the measure of another magnitude by k is increased k-fold. The ratio of the possible error to the whole

1 measure is as before, viz.

m

If now the number k is fractional and we wish to find an integral number of units in our product, we need not calculate

1

k more closely than to within

of unity. For, by the same

m

1

reasoning as above, provided the error in k is less than

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m

error in mk will be less than 1.

Example 1. The side of a square is, to within an inch, 57 yd. 2 ft. 3 in. Find the diagonal and assign the accuracy of the result.

57 yd. 2 ft. 3 in.=2079 in.
.., by Euc. I. 47, (diagonal)2=(2079 in.)2+(2079 in.)2

=2 (2079)2 sq. in.

.. diagonal=/2 x 2079 in. Now 2=1.4142 &c. This differs from 1.414 by less than •0003, i.e. less than gobo.

.. in finding an integral number of inches in the diagonal, we may use 1.414 for 12. .: diagonal=1'414 x 2079 in.=2939.706 in.

= 2940 in. or 81 yd. 2 ft. nearly. Since the original possible error was 1 inch, the possible error in the result is 12 inches, i.e. about 1} in.

Example 2. The ratio of the circumference to the diameter of a circle being 3:14159 &c., find the circumference of a circle whose diameter is, to within an inch, 13 yd. 8 in.

13 yd. 8 in.=476 in. Now 3:14159 &c. differs from 37 by less than ·002 or ado,

.. in finding an integral number of inches in the circumference we may use 37 instead of 3.141 &c.

.. circumference=2 of 476 in.=41 yd. 1 ft. 8 in. The result is correct to within about 3 inches.

Fundamental Proposition on Incommensurables. 35. Let there be two quantities which vary together, in such a way that to any value of one there corresponds a value of the other, and that when either increases the other increases. Then, if the ratio of any two commensurable values of the one is equal to the ratio of the two corresponding values of the other, the same will hold for all pairs of values whether commensurable or incommensurable.

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