(8) The base of a triangular field is 11.313708 chains find the length of the line parallel to the base, which divides it into two equal parts.

Answer: 8 chains.

(9) A square field contains 40 acres : determine the number of chains in its circuit. Find the area of a field of the same circuit, whose breadth is ths of its length.

Answers: 80ch. and 33ac. 2ro. 16po.

(10) A foot-path of uniform breadth around a circular field contains as much as ths of the field: compare the breadth of the path with the radius of the field. Answer: 1 : 4.

IMPERIAL WEIGHTS AND MEASURES.

220. The Weights and Measures made use of in this Kingdom having from length of time become subject to certain irregularities, in addition to the want of uniformity which generally prevailed in them, and the Standard Weights and Measures being at best but vaguely defined, the subject was at length laid before a Board of Commissioners: and, in accordance with a report furnished by them, An Act of Parliament which came into operation on the first of January 1826 was passed, establishing an uniform System of Imperial Weights and Measures, the leading features of which we shall now endeavour to place before the reader.

221. DEF. The Standard of Lineal Measure is a rod or beam whose length is called a yard which is equivalent to 3 feet or 36 inches: and the Standard Square and Cubic, Measures will therefore depend entirely upon it, as has been seen in the preceding pages.

At present we have no means of ascertaining why this particular length was originally fixed upon; but as it is most essential that it should always remain the same, it will be found convenient to refer it to something else which we have no reason to suppose ever undergoes any change.

Now the length of a Pendulum vibrating seconds or performing 86400 oscillations in the interval between

the Sun's leaving the meridian of a place and returning to it again, is always the same at a fixed place and under the same circumstances: and if this length be divided into 391392 equal parts, the yard is defined to be 360000 of these parts: also, conversely, since a yard is equal to 36 inches, it follows that the length of the seconds' pendulum expressed in inches is 39.1392.

The pendulum referred to in this country is one vibrating seconds at Greenwich or in London at the level of the sea in a non-resisting medium: and should the standard yard at any time be lost, it would be easy to have recourse to experiment for its recovery.

The standard yard being the general Unit of lineal measure, it follows that all lengths less than a yard will be expressed by fractions: and it is on this account that a lineal Inch or ten thousand of the aforesaid portions of the pendulum is conveniently adopted as the unit of lineal measure when applied to small magnitudes.

Hence also, the standard superficial and solid measures will be accurately ascertained and kept correct.

222. DEF. The Imperial Gallon is the standard unit of the measure of capacity, and is defined to be 277.274 cubic inches, the lineal inch being that above determined. The gallon and its multiples and parts are used to measure liquids, as Water, Spirits, &c., and dry goods, as Malt, Corn, &c., and the system is termed The Imperial Liquor and Dry Measure.

The Imperial Bushel consisting of eight gallons will consequently be 2218.192 cubic inches, and the form of the measure is defined by Act of Parliament: it is to be an upright cylinder whose internal diameter is 18.789 inches and depth 8 inches: but this can be of no importance when heaped measures are abolished.

The Act of Parliament directs that the Heaped Imperial Bushel shall be an upright cylinder whose diameter is not less than twice its depth, and that the height of the conical heap shall be at least three fourths of the depth, the boundary of its base being the outside of the measure: also, in heaped measure it is ordered that

3 Bushels make 1 Sack,

12 Sacks............1 Chaldron :

and the bushel is to be equal to 2815.4887 cubic inches.

223. DEF. The Imperial Pound Avoirdupois, which is the standard unit by means of which all heavy goods of large masses are weighed is defined to be the weight of one tenth part of an imperial gallon or of 27.7274 cubic inches of distilled water, ascertained at a time when the barometer stands at 30° and the height of Fahrenheit's thermometer is 62o.

If the weight of a cubic inch of distilled water be divided into 505 equal parts and each of such parts be defined to be a Half Grain, it follows that 27.7274 cubic inches contain very nearly 7000 such grains; and it is declared by Act of Parliament that 7000 Grains exactly shall be considered as the Pound Avoirdupois, so that according to the Table, 1oz. = 4374 grs. and 1dr. = 27}} grs.

A cubic foot of pure water, at a temperature 60o, weighs very nearly 1000 ounces; and for practical purposes, this fact should be remembered.

This weight receives its name from Avoirs the ancient name for goods or chattels and Poids signifying_weight, in the language of the country at the time of the Normans.

224. DEF. The Imperial Pound Troy is defined to be 5760 grains determined as in the last Article: and this weight, we are told in the "Report of the Commissioners of Weights and Measures," is retained, because all the Coinage has been uniformly regulated by it and all Medical Prescriptions or Formula now are and always have been estimated by Troy Weight, under a peculiar subdivision which the College of Physicians are most anxious to preserve.

It is also stated that there are reasons to believe that the word Troy has been derived from the Monkish name of Troy-novant, given to London, founded on the Legend of Brute, which is mentioned by some of the old English writers. The story avers that Brute a lineal descendant of Æneas about the year of the world 2855 founded the city of London, then called Trinovantum, which was afterwards corrupted into Trenovant or Troy-novant.

225. Hence, 1lb. avoirdupois : 1lb. troy: 7000:

5760, or as 175: 144; thus, 1lb. avoirdupois

175

lbs.

144

troy = 14oz. 11dwts. 16grs. troy, and 1lb. troy

175

avoirdupois 13 oz. 211 drs.: and the sub-divisions of each pound are easily compared.

226. DEF. The Unit of Angular Measure is called a Degree and is written or marked 1o: thus,

if the right angle BAC be supposed to be divided into 90 equal parts, each of them is called a degree and is considered to be the angular unit: and if the angle BAP be equal to any assigned number of such parts, as for instance 45, this angle will be 45° on the same scale that the right angle BAC is 90°: and thus the magnitudes of two or more angles may be compared.

If with the angular point A as a centre and any assumed radius AB, a circle be described and the diameters BAD, CAE be drawn at right angles to each other, dividing the circumference into four equal parts, BC, CD, DE, EB called Quadrants; then by Euclid vi. 33, the arcs BC, BP have the same ratio to each other as the angles BAC, BAP which they respectively subtend; and it is on this principle that a quadrantal arc is said to contain 90 degrees and therefore the whole circumference to be equal to 360 degrees; and although an angle and an arc being heterogeneous magnitudes cannot have the same unit of measure, it is clear that the division of the right angle into 90 equal portions of angular measure will correspond to the division of the quadrantal arc into 90 equal portions of lineal measure.

Thus, an angle and its subtending arc being proportional to each other will be connected by an invariable factor when the radius is given, so that either of them being known, the other may be ascertained.

That is, if the radius of the circle be the lineal unit, the circumference = 6.28318, nearly, by Article (211):

and therefore the quadrantal arc = 1.57079, nearly: whence, we have ‹ BAC : ▲ BAP :: arc BC: arc BP,

L

= 57°.2957795 x arc BP, nearly:

and therefore the magnitude of the

BAP is found by

multiplying 57°.2957795 by the number of lineal units contained in the corresponding arc:

also, 57°.2957795 = 57° . 17′ . 44′′. 48′′, nearly,

must consequently be regarded as the unit of angular measure in all cases where an angle is to be determined by means of the number of lineal units in its arc, or, which is the same thing, by the number of radii to which its arc is considered equivalent.

THE CALENDAR.

227. DEF. The interval of time between two passages of the Sun across the Meridian of any place when taken at its mean magnitude, is termed a Day or a Mean Solar Day, which is supposed to be divided into 24 equal portions called Mean Solar Hours. It appears from the observations and calculations of Astronomers that the time between the Sun's leaving a fixed point in his path called the Ecliptic and returning to it again, consists of 365.242264 such days or of 365 days, 5 hours, 48 minutes, 51 seconds, very nearly, which is therefore termed a Solar Year: and thus the solar year, as here defined, does not consist of an exact number of solar days but is always expressed in days by a fractional quantity.

For the purposes of civil life it would be exceedingly inconvenient that one year should commence at one time of the day and another at a different time: and this circumstance gave rise to the invention of the Civil Year which the next three Articles will explain.