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 involves the use of geometrical theorems, but it is not concerned with problems of geometrical construction.
Mensuration of the Circle.
For the subjects under this heading see the articles CONIC SECTIONS; CIRCLE; CURVE; GEOMETRICAL CONTINUITY; GEOMETRY, Axioms of; GEOMETRY, Euclidean; GEOMETRY, Projective; GEOMETRY, Analytical; GEOMETRY, Line; KNOTS, MATHEMATICAL THEORY OF; MENSURATION; MODELS; PROJECTION; Surface; Trigonometry.
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.
(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.
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.
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."
Philosophy, grammar, the history and theory of language, rhetoric, law, arithmetic, astronomy, geometry, mensuration, agriculture, naval tactics, were all represented.
From these results the mensuration of any figure bounded by circular arcs and straight lines can be determined, e.g.
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.
Pierpoint's Mensuration Formulae (1902) is a handy collection.
C. Turner, Graphics applied to Arithmetic, Mensuration and Statics (1907).
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.
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.