It is, for instance, practically impossible to obtain reliable evidence as to the regularity of employment in any industry in the 17th century, and the best approximations and devices we can invent are very poor substitutes for what we really want.
Amongst insects with imperfect metamorphosis the nearest approximations to the true pupa of the Holometabola are to be found in the subimago a From Chittenden, Bull.
The problem of finding a radius vector satisfying this condition is one which can be solved only by successive approximations, or tentatively.
A remarkable feature is the bare statement of a number of very close approximations to the square roots of numbers which are not complete squares.
Hero's expressions for the areas of regular polygons of from 5 to 12 sides in terms of the squares of the sides show interesting approximations to the values of trigonometrical ratios.
As far as the order to which he carried the approximations - which, however, were based on a simplifying hypothesis that the molecules influenced each other through mutual repulsions inversely as the fifth power of their distance apart--the result was that the equations of motion of the gas, considered as subject to viscous and thermal stresses, could be satisfied by a state of equilibrium under a modified internal pressure equal in all directions.
Rough approximations must always precede accurate measurements; and these returns, while often inaccurate, are better than nothing, and probably improve with each decade.
If we use c 1 to represent the chord of the whole arc, c 2 the chord of half the arc, and c 4 the chord of one quarter of the arc, then corresponding to (i) and (iii) of § 70 or § 79 we have a (8c 2 - c i) and4 5 (256c 4 - 40c2+ci) as approximations to the length of the arc. The first of these is Huygens's rule.
Such theories, as we have noticed, are extremely likely to be only approximations in ancient times, even if recognized then; and our data are quite inadequate for clearing the subject.
For the exact determination of the last element the census affords no precise data, but affords material for various approximations, based either upon the elimination of the probable progeny of immigrants since 1790; on the known increase of the whites of the South, where the foreign element has always been relatively insignificant; on the percentage of natives having native grandfathers in Massachusetts in 1905; or upon the assumed continuance through the 19th century of the rate of native growth (one-third decennially) known to have prevailed down at least to 1820.
The following table gives the values of this constant and several expressions involving it Useful fractional approximations are 22/7 and 355/113.
Pappus gives several solutions of this problem, including a method of making successive approximations to the solution, the significance of which he apparently failed to appreciate; he adds his own solution of the more general problem of finding geometrically the side of a cube whose content is in any given ratio to that of a given one.
In actual practice, however, decimals only represent approximations, and the process has to be modified (§ 111).
When a first approximation has been obtained in this way, further approximations can be obtained in various ways.
In theory these successive approximations may be carried as far as we please, but in practice the labour of executing each approximation is so great that we are obliged to stop when the solution is so near the truth that the outstanding error is less than that of the best observations.