2. The gable ends of a barn are each 28 feet wide, and the per pendicular height of the ridge above the eaves is 7 feet; how many feet of boards will be required to board up both gables ?

CASE V.

Ans. 196 feet.

476. To find the circumference or the diameter of a circle. 477. A Circle is a figure bounded by one uniform curved line.

478. The Circumference of a circle is the curved line bounding it.

479. The Diameter of a circle is a straight line passing through the center, and terminating in the circumference.

It is proved in geometry that in every circle the ratio between the diameter and the circumference is 3.1416+. Hence the

RULE. I. To find the circumference. — Multiply the diameter by 3.1416.

II. To find the diameter.

- Multiply the circumference by .3183.

EXAMPLES FOR PRACTICE.

1. What length of tire will it take to band a carriage wheel 5 feet in diameter ?

Ans. 15 ft. 8.4 in.

2. What is the circumference of a circular lake 721 rods in diameter ? Ans. 7 mi. 25 rds. 1.54 + ft. 3. What is the diameter of a circle 33 yards in circumference? Ans. 10.5 yards.

CASE VI.

480. To find the area of a circle.

From the principles of geometry is derived the following

RULE. I. When both diameter and circumference are given; Multiply the diameter by the circumference, and divide the product by 4.

II. When the diameter is given;- Multiply the square of the diameter by .7854.

III. When the circumference is given; — Multiply the square of the circumference by .07958.

EXAMPLES FOR PRACTICE.

1. The diameter of a circle is 113, and the circumference 355; what is the area? Ans. 10028.75. 2. What is the diameter of a circular island containing 1 square mile of land? Ans. 1 mi. 41 rd. 1.4+ft. 3. A man has a circular garden requiring 84 rods of fencing to inclose it; how much land in the garden? Ans. 3 A. 81.5+ P.

THE METRIC SYSTEM

OF

WEIGHTS AND MEASURES.*

INTRODUCTION.

The metric system of weights and measures

so called, because the metre is the unit from which the other units of the system are derived-had its origin in France during the Revolution, a time when all regard for institutions of the past was repudiated. In the year 1790, the French government resolved to introduce a new system; and, in order that it might be received with general favor, other countries were invited to join with it in the choice of new units. In response to this invitation, a large number of scientific men, commissioned by various countries, met in Paris, in consultation with the principal men of France. In the year 1791, a commission, nominated by the Academy of Sciences, was appointed by the Government to prepare the new system. The first work of the commission was to select a standard of lengths from which the system of units adopted might at any time be restored if from any cause the original unit should be lost. A quadrant of the earth's meridian was chosen as the standard, and the ten-millionth part of it taken as the unit of lengths, which was called a metre. In 1795, this standard and a provisional metre whose length was determined from measurements

* M. MCVICAR, A. M., Principal of the State Normal and Training School at Brockport, N.Y., a most thorough and critical scholar as well as teacher, prepared this article, which contains many practical improvements in Notation, Nomenclature, and Applications, not before presented to the public.

Entered, according to Act of Congress, in the year 1867, by D. W. FISH, A. M., in the Clerk's Office of the District Court of the United States for the Southern District of New York

of the earth's meridian, which had already been made, was adopted by the government.

In the meantime, two eminent astronomers, Mechain and Delambre, were engaged in determining the exact length of the arc of the meridian between Dunkirk in the north of France, and Barcelona in Spain. At a later period, Biot and Arago measured the prolongation of the same meridian as far as the island of Formentara. From these measurements, together with one formerly made in Peru, they deduced, as they supposed, the exact distance from the equator to the pole, which differed slightly from the standard assumed in 1795. In 1799, a law was passed changing the length of the metre adopted in 1795 so as to conform with this difference. The metre thus determined was marked by two very fine parallel lines drawn on a pla tinum bar, and deposited for preservation in the national archives.

While a part of the commission were engaged in establishing the exact length of the metre, other members pursued a course of investigation for the purpose of determining a unit of weights, which would sustain an invariable relation to the unit of lengths. As the result of their investigations, the weight of a cube of pure water whose edge was one-hundredth part of a metre was the unit chosen. The water was weighed in a vacuum, at a temperature of 4° C., or 39.2° F., which was supposed to be the temperature of greatest density. This weight was called a gramme; and a piece of platinum weighing one thousand grammes was deposited as the standard of weights in the national archives.

Had the work of the commission ended in determining these standards of lengths and weights, their labor would have been futile. For, while the conception of basing their system upon an absolute standard in nature was good, the execution proved a failure. Later investigations have shown that the metre is less than the ten-millionth part of the earth's meridian; consequently the metric system of weights and measures is referable not to an invariable standard in nature, but to the platinum metre deposited in the national archives of France. The great benefits which result from the labors of the commission arise from the adoption of the decimal scale of units, and a simple yet general and expressive nomenclature. The amount of

time and money used in carrying on exchanges between different countries, which would be saved by the universal adoption of this system, is incalculable. The system was declared obligatory throughout the whole of France after Nov. 2, 1801; but, owing to the prejudices of the people in favor of established customs, and the confusion consequent upon the use of the new measures, the Government, in 1812, adopted a compromise, in the système usuel, whose principal units were the new ones, while the divisions and names were nearly those formerly in use, ascending commonly in the ratios of two, three, four, eight, or twelve. In 1837, the government abolished this system, and enacted a law attaching a penalty to the use of any other than the metric system after Jan. 1, 1841. Since that time, the system has been adopted by Spain, Belgium, and Portugal, to the exclusion of other weights and measures. In Holland, other weights are used only in compounding medicines. In 1864, the system was legalized in Great Britain; and its use, either as a whole or in some of its parts, has been authorized in Greece, Italy, Norway, Sweden, Mexico, Guatemala, Venezuela, Ecuador, United States of Columbia, Brazil, Chili, San Salvador, and Argentine Republic. In 1866, Congress authorized the metric system in the United States by passing the following bills and resolution :

AN ACT гO AUTHORIZE THE USE OF THE METRIC SYSTEM OF WEIGHTS AND MEASURES.

Be it enacted by the Senate and House of Representatives of the United States of America in Congress assembled, That, from and after the passage of this Act, it shall be lawful throughout the United States of America to employ the Weights and Measures of the Metric System; and no contract or dealing, or pleading in any court, shall be deemed invalid, or liable to objection, because the weights or measures expressed or referred to therein are weights or measures of the Metric System.

SECTION 2. And be it further enacted, That the tables in the schedule hereto annexed shall be recognized in the construction of contracts, and in all legal proceedings, as establishing, in terms of the weights and measures now in use in the United States, the equivalents of the weights and measures expressed therein in terms of the Metric System; and said tables may be lawfully used for computing, determining, and expressing in customary weights and measures, the weights and measures of the Metric System.

A BILL TO AUTHORIZE THE USE IN POST OFFICES OF THE WEIGHTS OF THE DENOMINATION OF GRAMMES.

Be it enacted by the Senate and House of Representatives of the United States of America in Congress assembled, That the Postmaster General be, and he is hereby, authorized and directed to furnish to the post-offices exchanging mails with foreign countries, and to such other offices as he shall think expedient, postal balances denominated in grammes of the metric system; and, until otherwise provided by law, one-half ounce avoirdupois shall be deemed and taken for postal purposes as the equivalent of fifteen grammes of the metric weights, and so adopted in progression; and the rates of postage shall be applied accordingly.

JOINT RESOLUTION TO ENABLE THE SECRETARY OF THE TREASURY TO FURNISH TO EACH STATE ONE SET OF THE STANDARD WEIGHTS AND MEASURES OF THE METRIC SYSTEM.

Be it resolved by the Senate and House of Representatives of the United States of America in Congress assembled, That the Secretary of the Treasury be, and he is hereby, authorized and directed to furnish to each State, to be delivered to the governor thereof, one set of the standard weights and measures of the metric system, for the use of the States respectively.