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MENSURATION

OF

LINES, SURFACES, AND VOLUMES.

BY

DAVID MUNN, F. R.S. E.

MATHEMATICAL MASTER, ROYAL HIGH SCHOOL OF EDINBURGH;
AUTHOR OF THEORY OF ARITHMETIC,' ETC.

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Edinburgh:

Printed by W. and R. Chambers.

PREFACE.

THE following pages have been transcribed from notes made by the author from time to time in teaching Mensuration in his own classes, and are intended for those pupils only who have previously acquired some knowledge of Geometry and Algebra. Indeed it is not until a pupil has acquired this knowledge that he can take up the subject with any degree of intelligence, or derive any educational advantage from its study.

The subject, however, is so important, and its results are of so much practical utility, that it is not unusual to supplement the ordinary course of arithmetic by the addition of rules and exercises on Mensuration. These rules the arithmetical pupil may follow, and the exercises he may work; but beyond taxing his memory and exercising his fingers, they can be of little or no advantage to him. On the other hand, when he enters upon the study of Mensuration with a knowledge of Geometry and Algebra, he can frame and investigate rules for himself, and thus render the subject an efficient mears

of mental discipline. Besides, the results of Mensuration, so important in themselves, afford the student a ready means of illustrating and interpreting geometrical processes and algebraical formulæ.

The author has to express his obligations to Mr J. J. A. BLACK, his able assistant, who has carefully wrought out the exercises proposed, and added his results in the form of answers at the end of the book.

ROYAL HIGH SCHOOL, EDINBURGH, March 1873.

PROPOSITION

CONTENT S.

PART I.-LINES.

PAGE.

I. Having given any two sides of a right-angled triangle,
to find the remaining side.........................
II. Given the three sides of a triangle, to find the length
of the perpendicular on any of the sides from the
opposite angle......

III. Given the three sides of a triangle, to find the length
of the line drawn from any of the angles to the
bisection of the opposite side......

IV. Given the three sides of a triangle, to find the length
of the line bisecting any of the angles.........

V. Given the three sides of a triangle, to find the
diameter of the circumscribing circle...

VI. To divide a line into extreme and mean ratio...

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VII. Having given the height of an arc and the radius of a circle, to find the length of the chord.........

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VIII. Having given the side of a regular polygon inscribed
in a given circle, to find the side of the regular
polygon inscribed with double the number of sides. 19
IX. Having given the side of a regular polygon inscribed
in a given circle, to find the side of the regular
polygon of the same number of sides circumscribing
the circle.

X. Given the perimeter of a regular polygon inscribed
within a circle, and the perimeter of a similar cir-
cumscribed polygon, to find the perimeters of the
regular inscribed and circumscribed polygons of
double the number of sides......

XI. To find the side of a regular decagon inscribed in a

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circle........

EXERCISES (2).

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