MENSORES

MENSORES, amongst the Romans, were harbingers, whose business it was to go before the emperor, and fix upon lodgings for him when he travelled into any of the provinces. They also marked out encampments, and assigned every regiment its post.

Mensores were likewise land-surveyors, architects, or appraisers of houses and public buildings. The distributors of provisions in the army were called mensores frumentarii; and the appellation of mensores was also applied to servants who waited at table.

EVERY branch of the mathematics which has for its object the comparison of geometrical quantities, and the determination of their proportions to each other, may be comprehended under the general name Mensuration. So that, taking the term in its most extensive sense, whatever is delivered in this work under the titles GEOMETRY, TRIGONOMETRY, CONIC SECTIONS, part of ALGEBRA, and a very considerable portion of FLUXIONS, may be considered as constituting particular branches of this general theory.

The term mensuration, however, is also frequently used in a less extensive sense, and is applied to a system of rules and methods by which numerical measures of geometrical quantities are obtained. And it is to this limited view of the subject that we propose to confine our attention in the present treatise. In general, it will only be necessary to

give the practical rules, as we have already explained their foundation when treating of GEOMETRY, CONIC SECTIONS, and FLUXIONS; but, in addition to the rules, in a few instances, we shall give their demonstrations.

In all practical applications of mathematics it is necessary to express magnitudes of every kind by numbers. For this purpose a line of some determinate length, as one inch, one foot, &c., is assumed as the measuring unit of lines, and the number expressing how often this unit is contained in any line, is the numerical value or measure of that line.

A surface of some determinate figure and magnitude is assumed as the measuring unit of surfaces, and the number of units contained in any surface is the numerical measure of that surface, and is called its area. It is usual to as-

Table of measures. sume, as the measuring unit of surfaces, a square, whose side is the measuring unit of lines.

A solid of a determinate figure and magnitude is in like manner assumed as the measuring unit of solids, and the number of units contained in any solid is its solidity or content. The unit of solids is a cube, each of whose edges is the measuring unit of lines, and consequently each of its faces the measuring unit of surfaces.

A right angle is conceived to be divided into ninety equal angles; and one of these, called an angle of one degree, is assumed as the measuring unit of angles.

The measures generally employed in the application of mensuration to the common affairs of life, and their proportions to each other, are expressed in the following tables:

These measures were recognised and established by an act of Parliament in 1825; but they were the measures in common use before that time. The length of the standard yard is the distance between two points marked on a straight rod of brass commonly called Bird's Standard Yard of 1760. The length of the yard is to that of a pendulum vibrating seconds of mean time in the latitude of London, in a vacuum at the level of the sea, as 36 to 39.1393, the temperature of the brass rod being 62° of Fahrenheit's thermometer.

Derived from the Scottish Standard Ell, in the custody of the town of Edinburgh.

Note. We have given these Scottish measures for the purpose of comparison with Imperial measure, by which the Scottish measures, being now set aside, will be gradually superseded.

Note 1. To convert Scotch acres into Imperial, to the Right lines number of Scotch acres add its $\backslash frac\{1\}\{4\}$th, also its $\backslash frac\{1\}\{8\}$th; and in addition to these, if accuracy be required, the $\backslash frac\{1\}\{16\}$th part of the $\backslash frac\{1\}\{8\}$th. Suppose, for example, 100,000 Scotch acres are to be converted into Imperial acres:

2. The same rule will apply to the conversion of rent or value, and produce of Scotch acres.

The Imperial gallon is the space occupied by 10 lbs. of water, weighing 70,000 grains.

The Imperial ale gallon is now the same as the Imperial wine gallon.

The Winchester bushel measure is a cylinder 18$\backslash frac\{1\}\{2\}$ inches wide, and eight inches deep, and contains 2150.4252 cubic inches.

The Imperial bushel contains 2218.191 cubic inches.

The Standard Pint is the Stirling Jug, which contains 104.2034 cubic inches, and holds 26,306.982 grains of distilled water, at 62° of Fahrenheit's thermometer, the height of the barometer being 30 inches.

The Scotch Gallon = 3.0065122 Imperial Gallons.

The Linlithgow Wheat Firlet is to the Imperial Standard Bushel as 0.998256 to 1. This is not far from a ratio of equality, so that the Scotch Standard Wheat Boll, Firlet, Peck, and Lippie nearly correspond with the Imperial Coomb and its submultiples, the Bushel, Peck, and Pottle.

The Linlithgow Barley Firlet is to the Imperial Standard Bushel as 1.4562794 to 1.

The rules by which certain of the sides or angles of a triangle are to be found, when other sides and angles are

Right lines given, might be considered as belonging to this part of mensuration. But as these are fully investigated and explained in the article PLANE TRIGONOMETRY, it is not necessary to deliver them also here. Referring therefore to that article, we shall employ the remainder of this section in the application of trigonometry to the mensuration of heights and distances.