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16. We will suppose that the length thus measured is the side of a square, and that the workman wishes to know to the same degree of accuracy as his measurement, what is the length of the diameter of the square.
He can find it thus.
The diameter of a square=12 x its side, .: the diameter of this square=12 x 891 thirty-second parts
of an inch.
= 1259.8. .. the required diameter = 1260 thirty-second parts of
an inch, nearly. The error being less than one thirty-second part of an inch.
17. The student will be able to see from the above example, that if the value of an incommensurable number is found to 4 figures, a very considerable degree of accuracy is attained. Also that a much greater degree of accuracy is attained for every additional figure.
18. It is no advantage in calculations such as the above to have the value of such quantities as /2 calculated to any greater degree of accuracy than the observed measurement. For instance, our calculation being correct as far as the whole numbers are concerned we should gain nothing by using 1.4142 instead of 1.414 for 12. This would give us the answer 1260.0 instead of 1259.8; the difference being a fifth of a thirty-second part of an inch, a quantity by hypothesis too small to be of any importance.
Example 1. The side of an equilateral triangle contains 2 feet; find, correct to the ten-thousandth part of a foot, tho length of the perpendicular drawn from an angular point to the opposite side.
Here (as in Example 2, p. 6), let the perpendicular contain x feet, then
x2 sq. ft. = 22 sq. ft. – 12 sq. ft.,
Example 2. Find the length, correct to the ten-thousandth part of a foot, of the side of the square described upon a diameter whose length is a feet.
Let x be the number of feet in each of the sides of the square. Then
x2 sq. ft. + x* sq. ft. =a? sq. ft.
..x2 + x2=a?,
.: 2x2=a?, .:2.x=a,
a 2 72 2
*(1:4142)=ax •7071..., .. the length of the perpendicular=ax •7071...feet.
* EXAMPLES. IV.
(1) Find, correct to the thousandth part of a foot, the length of the diameter of a square whose side is 7 feet.
(2) Find, correct to a yard, the length of the diameter of a square whose side is one mile..
(3) Find, correct to the hundredth part of an inch, the hypotenuse of a right-angled triangle whose sides are 3 ft. 61 in. and 3 ft. 4 in. respectively.
(4) Find, to the nearest inch, the side of a square whose area is 1000 square yards.
(5) Find, correct to the tenth part of a foot, the diameter of a square field whose area is ten acres.
(6) Find, to the nearest inch, the side of a square field whose area is one acre.
(7) Find the height of an equilateral triangle whose side is 10 feet.
(8) Find the height of an equilateral triangle whose side is 5:32 feet.
(9) The top of a table measures 24:6 inches and 41:3 inches along two adjacent sides. What should the diameter measure if the table is rectangular ?
(10) Find, to the nearest inch, the diameter of a lawn-tennis court whose length is 78 feet, and breadth 36 feet, supposing that it is properly marked out.
ON THE RELATION BETWEEN THE CIRCUMFERENCE OF A
CIRCLE AND ITS DIAMETER.
19. THE circumference of a circle is a line, and therefore it has length.
We might imagine the circumference of a circle to consist of a flexible wire ; if the circular wire were cut at one point and straightened, we should have a straight line of the same length as the circumference of the circle.
20. A polygon is a figure enclosed by any number of straight lines.
21. A regular polygon has all its sides equal and all its angleg equal.
22. The perimeter of a polygon is the sum of its sides.
23. If we have two circles in which the diameter of the first is greater than the diameter of the second, it is evident that the circumference of the first will be greater than the circumference of the second.
24. It seems, therefore, not unlikely that, if the diameter of the first circle be twice that of the second, the circumference of the first will be twice that of the second.
25. And also not unlikely that whatever be the ratio of the circumference of the first circle to its diameter, the same will be the ratio of the circumference of the second circle to its diameter,
This suggests that it is not unlikely that the circumference of a circle = k times its diameter, where k is some number which is the same for all circles.
We shall presently prove that this is the case.
26. But although we can prove that the circumference of a circle
= a fixed numerical quantity, its diameter the method of calculating the value of this number is beyond the limits of an elementary treatise.
27. We shall therefore simply state here, what is proved in the Higher Trigonometry,
(i) that this numerical value is incommensurable,
28. When we say that this number is incommensurable we mean (cf. Chapter II.) that its exact value cannot be stated as an arithmetical fraction.
It also happens that we have no short algebraical expression such as a surd, or combination of surds, which represents it exactly.
So that we have no numerical expression whatever, arithmetical nor algebraical, to represent exactly the ratio of the circumference of a circle to its diameter.
Hence the universal custom has arisen, of denoting its exact value by the letter 1.