History

of science,

Miles' 8-2-23

9253

[v]

PREFACE.

I

N the following Treatife are laid down the Principles for the Menfuration of Magnitude, with fuch other Parts of the Mathematicks only, as are neceffarily leading thereto, and their Application in the Investigating and Demonftrating the feveral Theorems for. meafuring those plain Surfaces and Solids, which more immediately belong to Gauging: The whole being originally defign'd for the benefit of the Officers of the Excife, that they might not only be acquainted with the most eafy and exact Methods for Practice, but be at the fame time certain of the Truth of the Rules made use of.

Care has been taken to avoid meddling with the Methods practifed by fome others; or any way to be concerned in refuting their Errors; but here the only business is, to give the neceffary Rules for the just and ready Practice of the Art of Gauging; fhewing the Manner of coming at thofe Rules, from Principles within themfelves the most perfect of all human Knowledge.

For that End, it is hoped, every thing here treated of, is explain'd in fo eafy a manner, as to

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be

be within the reach even of Beginners, that
will give that Attention which is unavoidably
required in the Study of every Part of the Ma-
thematicks: And it may with Jafety be affirm'd,
that without a competent Skill in Algebra and
Geometry, it is abfolutely impoffible for any Per-
fon to determine, whether the Rules given by
common Writers upon this Subject be true or false,
and much less to make any (even the least) Im-
provement in this Part of Science.

Tho the Practice of Gauging may be al
lowed to be a fufficient Qualification for eve-
ry Officer: Yet, 'tis well known from common
Experience, that, by far the greatest Part of
them, neither want Abilities to understand, or
Ambition to have a reasonable Proof of the
Truth of the Rules and Methods on which the
Practice is founded.

And as no Writer hitherto has thought fit
to give both the Theory and Practice of Gaug-
ing in one and the fame Book, I hope that may
be look'd on as a fufficient Apology for my Un-
dertaking to do what I here offer, which is di-
vided into three Parts.

The first, as an Introduction to the other
Parts, containing the Method of computing de-
cimally, by the Pen, with all the ufeful Con-
tractions: And by the Sliding-Rule with its
Conftructions, and a very full and particular
Account of its Ufe, together with the Reafon of
the feveral Operations made by each Method.

In

In the fecond Part, there are fome of the primary Principles of plain Geometry, with thofe of the Conic Sections, as preparatory to what follows: Where fuch general Propofitions, for the Meafuring of Plane Surfaces and So lids, are laid down, that the Rules for most of the common ones, are deduced by way of Corollaries.

You have here particular Rules for Menfuration of all right-lined Plane Surfaces, Cir cles, Conic Sections, and for Solids, fuch as Paral lelopipedons, Prifms, Conoids, Spindles, with the Segments and Fruftums of thefe: Some of thefe perhaps were not published before, fuch I fuppofe, are the Series for the Circle (in Pag. 100.). for the Hyperbola (in Pag. 109.) and the Theo rems for measuring the Circular, Elliptic, and Hyperbolic Spindles (in Pag. 156.) Then are given the Investigation of Theorems, for reducing thefe Solids nearly into Cylinders of the fame Length: From whence is deduced, occafionally, the common fixed Multipliers, and fhewn bow far they may be depended upon, in the Practice of Gauging. Next, a general Theorem is given, for the Menfuration of all Pyramidical or Conic Hoofs, and the fame is accomplish'd in the three Hoofs of an upright Cone, and that of the Square Pyramid: On this depends the method of Gauging an inclined Pyramidical Conical Tun.

Then the Method is fhewn of Measuring by Approximation, which, if brought into Ufe, would

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would render the Art of Gauging far more compleat, than it has been hitherto: For by this Method, the Measure of any Cafk, Tun, Cop per, or Still may be found within any defired Degree of Truth, and that by only taking a fufficient Number of Dimenfions, which will ever be less than what is required in the common me→ thods of gauging Coppers, and the Conclufions will be much nearer the Truth. But for Cafks, there is, befides the ufual Data, required only a Diameter in the Middle between the Head and the Bung; from which the Form of the Cafk, in fome Degree, is determined.

This perhaps may at first be look'd upon as a Defign to introduce Novelty in a Matter fufficiently confirm'd by Experience, and by that means be rejected as a fpeculative Nicety.

But if it be confidered that the prefent Prac tice of Gauging depends upon affigning fome known Form to the Tun or Cafk to be gauged 3 and without affuming that Form, we cannot make one Step towards computing its Meafure: So that if we are not exact in our Affumption, it is not poffible we should be exact in its Content. And it may be with truth affirm'd, that there never was, ftrictly speaking, a Cafk in the Shape of any one of thofe Solid Figures whofe Names they bear; fince it never could happen fo, unless perfectly by Accident: The Maker having no thought of defigning them as fuch; and befides, fhould a Gask approach nearly