This applies not only to the geometrical principles but also to the arithmetical principles, and it is therefore of importance, in the earlier stages, to keep geometry, mensuration and arithmetic in close association with one another; mensuration forming, in fact, the link between arithmetic and geometry.
All exact relations pertaining to the mensuration of the circle involve the ratio of the circumference to the diameter.
General aspects of the subject are considered under Mensuration; Vector Analysis; Infinitesimal Calculus.
This use of formulae for dealing with numbers, which express magnitudes in terms of units, constitutes the broad difference between mensuration and ordinary geometry, which knows nothing of units.
Mensuration of the Circle.
Such knowledge, he here maintains, is really mensuration of pleasures and pains, whereby the wise man avoids those mistaken under-estimates of future feelings in comparison with present which we commonly call " yielding to fear or desire."
In 1764 he published his first work, The Schoolmaster's Guide, or a Complete System of Practical Arithmetic, which in 1770 was followed by his Treatise on Mensuration both in Theory and Practice.
The term " mensuration " is therefore ordinarily restricted to the measurement of areas and volumes, and of certain simple curved lengths, such as the circumference of a circle.
Mensuration involves the use of geometrical theorems, but it is not concerned with problems of geometrical construction.
On the other hand, mensuration, in its practical aspect, is of importance for giving reality to the formulae themselves and to the principles on which they are based.
(ii) The very earliest stages of mensuration should be directly associated with simple arithmetical processes.
The next stage is geometrical mensuration, where geometrical methods are applied to determine the areas of plane rectilinear figures and the volumes of solids with plane faces.
The third stage is analytical mensuration, the essential feature of which is that account is taken of the manner in which a figure is generated.
Mensuration Of Specific Figures (Geometrical) 22.
- The mensuration of the circle is founded on the property that the areas of different circles are proportional to the squares on their diameters.
- For elementary mensuration the ellipse is to be regarded as obtained by projection of the circle, and the ellipsoid by projection of the sphere.
37 The mensuration of earthwork involves consideration of quadrilaterals whose dimensions are given by special data, and of prismoids whose sections are D such quadrilaterals.
[[Mensuration Of Graphs 38.]] (A) Preliminary.
To illustrate the importance of the mensuration of graphs, suppose that we require the average value of u with regard to x.
The processes which have to be performed in the mensuration of figures of this kind are in effect processes of integration; the distinction between mensuration and integration lies in the different natures of the data.
The province of mensuration is to express the final result of such an elimination in terms of the data, without the necessity of going through the intermediate processes.
In elementary geometry we deal with lines and curves, while in mensuration we deal with areas bounded by these lines or curves.
The circle, for instance, is regarded geometrically as a line described in a particular way, while from the point of view of mensuration it is a figure of a particular shape.
Similarly, analytical plane geometry deals with the curve described by a point moving in a particular way, while analytical plane mensuration deals with the figure generated by an ordinate moving so that its length varies in a particular manner depending on its position.
In the same way, in the case of a figure in three dimensions, analytical geometry is concerned with the form of the surface, while analytical mensuration is concerned with the figure as a whole.
(B) Mensuration of Graphs of Algebraical Functions.
(C) Mensuration of Graphs Generally.
The relation between the inaccuracy of the data and the additional inaccuracy due to substitution of another figure is similar to the relation between the inaccuracies in mensuration of a figure which is supposed to be of a given form (§ 20).
In the case of a trapezette, for instance, the data are the magnitudes of certain ordinates; the problem of interpolation is to determine the values of intermediate ordinates, while that of mensuration is to determine the area of the figure of which these are the ordinates.
- FOr applications of the prismoidal formula, see Alfred Lodge, Mensuration for Senior Students (1895).
Other works on elementary mensuration are G.
Chivers, Elementary Mensuration (1904); R.
Edwards, Elementary Plane and Solid Mensuration (1902); William H.
Pierpoint's Mensuration Formulae (1902) is a handy collection.
C. Turner, Graphics applied to Arithmetic, Mensuration and Statics (1907).
The mensuration of the cube, and its relations to other geometrical solids are treated in the article Polyhedron; in the same article are treated the Archimedean solids, the truncated and snubcube; reference should be made to the article Crystallography for its significance as a crystal form.
From these results the mensuration of any figure bounded by circular arcs and straight lines can be determined, e.g.
Philosophy, grammar, the history and theory of language, rhetoric, law, arithmetic, astronomy, geometry, mensuration, agriculture, naval tactics, were all represented.
Besides agriculture, the course of instruction at the college includes chemistry, natural and mechanical philosophy, natural history, mensuration, surveying and drawing, and other subjects of practical importance to the farmer, proficiency in which is tested by means of sessional examinations.
In actual practice, surds mainly arise out of mensuration; and we can then give an exact definition by graphical methods.
While mensuration is concerned with the representation of geometrical magnitudes by numbers, graphics is concerned with the representation of numerical quantities by geometrical figures, and particularly by lengths.
Mensuration is not concerned with the first of these two processes, which forms part of the art of measurement, but only with the second.
It is also convenient to regard as coming under mensuration the consideration of certain derived magnitudes, such as the moment of a plane figure with regard to a straight line in its plane, the calculation of w]iich involves formulae which are closely related to formulae for determining areas and volumes.
If these are included in the description " mensuration," the subject thus consists of two heterogeneous portions - elementary mensuration, comprising methods and results, and advanced mensuration, comprising certain results intended for practical application.
Mensuration, then, is mainly concerned with quadratureformulae and cubature formulae, and, to a not very clearly defined extent, with the methods of obtaining such formulae; a quadrature-formula being a formula for calculating the numerical representation of an area, and a cubature-formula being a formula for calculating the numerical representation of a volume, in terms, in each case, of the numerical representations of particular data which determine the area or the volume.
In developing a system of mensuration-formulae the importance of this latter group of cases must not be overlooked.
As a result of the importance both of the formulae obtained by elementary methods and of those which have involved the previous use of analysis, there is a tendency to dissociate the former, like the latter, from the methods by which they have been obtained, and to regard mensuration as consisting of those mathematical formulae which are concerned with the measurement of geometrical magnitudes (including lengths), or, in a slightly wider sense, as being the art of applying these formulae to specific cases.
The main object to be aimed at, therefore, in the study of elementary mensuration, is that the student should realize the possibility of the numerical expression of areas and volumes.
- The methods of mensuration fall for the most part under one or other of three main heads, viz.
Arithmetical mensuration, geometrical mensuration, and analytical mensuration.
The most elementary stage is arithmetical mensuration, which comprises the measurement of the areas of rectangles and parallelepipeds.
This may be introduced very early; square tablets being used for the mensuration of areas, and cubical blocks for the mensuration of volumes.
The measure of the area of a rectangle is thus presented as the product of the measures of the sides, and arithmetic and mensuration are developed concurrently.
There are also cases in which graphics and mensuration are used jointly; a variable numerical quantity is represented by a graph, and the principles of mensuration are then applied to determine related numerical quantities.
The statement that, if the adjacent sides of a rectangle are represented numerically by 3 and 4, the diagonal is represented by 5, is as much a matter of mensuration as the statement that the area is represented by 12.
Lambert, Computation and Mensuration (1907).