PART II.

GEOMETRY

EOMETRY is the Science of Extenfion, and is employ'd in the Confideration of Lines, Surfaces and Solids; as all Extenfion is distinguished into Length, Breadth, and Thickness.

This Science had its Rife among the Egyptians,

who were in a manner compelled to invent it, to Of its Origin. remedy the Confufion which generally happen'd

in their Lands, from the Overflowings of the River Nile, which carried away all Boundaries, and effaced all the Limits of their Poffeffions: And thus this Invention, which at firft confifted only in measuring the Lands, that every one might have what belonged to him, was called Land-meafuring, or Geometry; but the Egyptians afterwards applied themselves to more subtle Refearches, and from a very mechanical Exercife, infenfibly produced this fine Science, which deferves to be placed among thofe of the first Rank.

Geometry is not barely ufeful, but even abfo

lutely neceffary. It is by the Help of Geometry Of its Use. that Aftronomers make their Obfervations, re

gulate the Duration of Times, Seafons, Years, and Cycles,

and

and measure the Distance, Motion and Magnitudes of the Heavenly Bodies.

It is by Geometry that Geographers fhew us the Magnitude of the whole Earth, delineate the Extent of Seas, and the Divifions of Empires, Kingdoms, and Provinces.

It is from this Science that Architects derive their just Meafures in the Conftruction of public Edifices, as well as of private Houfes.

It is by its Affiftance that Engineers conduct all their Works, take the Situations and Plans of Towns, the Distance of Places, and in fine, the Measure of fuch things as are only acceffible to the Sight.

Such as are in the military Service, are obliged to apply themselves to this Science. It is not only an Introduction to Fortification, (which fhews them how to build Ramparts for the Defence of Places, and to conftruct and make Machines to deftroy them) but also gives them great Knowledge and Readiness in the military Art, in the drawing up an Army in Order of Battle, and in marking out the Ground in Encampments. It also fhews them how to make Maps of Countries, to take the Plans of Towns, Forts, and Caftles, to measure all kinds of Dimenfions acceffible or inacceffible, to give Designs, and in fine, to render themselves as ferviceable by their Understanding and Science, as by their Strength and Courage.

All who profefs Defigning fhould know fomething of Geometry, because they cannot otherwife perfectly understand Architecture nor Perspective, which are two things abfolutely neceffary in their Art.

Mufic, Mechanics, and, in a word, all the Sciences which confider Things fufceptible of more, and lefs; i. e. all the precife and accurate Sciences, may be referred to Geometry: for all fpeculative Truths confifting only in the Relations of Things, and in the Relations between thofe Relations, they may be all referred to Lines. Confequences may be drawn from them; and thefe Confequences, again, being rendered fenfible by Lines, they become permanent Objects, conftantly expofed to a rigorous Attention, and Examination: and thus we have infinite Opportunities both of inquiring into their Certainty, and purfuing them farther.

The Reafon, for instance, why we know fo diftinctly, and mark fo precifely, the Concords called Octave, Fifth, Fourth, c. is, that we have learnt to exprefs Sounds by Lines, i. e. by Chords accurately divided; and that we know that the

Chord,

Chord, which founds Octave, is double of that which it makes Octave withal; that the fifth is in the fefquialterate Ratio, or as three to two; and fo of the rest.

The Ear itself cannot judge of Sounds with fuch Precision; its Judgments are too faint, vague, and variable to form a Science. The fineft, beft tuned Ear, cannot diftinguish many of the Differences of Sounds; whence many Muficians deny any fuch Differences; as making their Senfe their Judge. Some, for inftance, admit no Difference between an Octave and three Ditones and others, none between the greater and leffer Tone; the Comma, which is the real Difference, is infenfible to them; and much more the Scifma, which is only half the Comma.

It is only by Reafon, then, that we learn, that the Length of the Chord which makes the Difference between certain Sounds, being divifible into feveral Parts, there may be a great Number of different Sounds contained therein, ufeful in Mufic, which yet the Ear cannot diftinguish. Whence it follows, that had it not been for Arithmetic and Geometry, we had had no fuch thing as regular, fixed Mufic; and that we could only have fucceeded in that Art by good Luck, or Force of Imagination, i. e. Mufic would not have been any Science founded on inconteftable Demonftrations: though we allow that the Tunes composed by Force of Genius and Imagination, are usually more agreeable to the Ear, than those compofed by Rule.

So, in Mechanics, the Heaviness of a Weight, and the Diftance of the Center of that Weight from the Fulcrum, or Point it is fuftained by, being fufceptible of plus and minus, they may both be expreffed by Lines; whence Geometry becomes applicable hereto; in virtue whereof, infinite Difcoveries have been made, of the utmost Use in Life.

Geometrical Lines and Figures, are not only proper to reprefent to the Imagination the Relations between Magnitudes, or between Things fufceptible of more and lefs; as Spaces, Times, Weights, Motions, &c. but they may even represent Things which the Mind can no otherwife conceive, e. gr. the Relations of incommenfurable Magnitudes.

We do not, however, pretend, that all Subjects Men may have occafion to enquire into, can be expreffed by Lines. There are many not reducible to any fuch Rule: thus, the Knowledge of an infinitely powerful, infinitely juft God, on whom all Things depend, and who would have all his Creatures execute his Orders, to become capable of being happy,

is the Principle of all Morality, from which a thousand undeniable Confequences may be drawn, and yet neither the Principle, nor the Confequences can be expreffed by Lines, or Figures. Malebr. Recher. de la Ver. T. ii.

Indeed, the ancient Egyptians, we read, used to express all their Philofophical, and Theological Notions by Geometrical Lines. In their Researches into the Reafon of things, they obferved, that God, and Nature, affect Perpendiculars, Parallels, Circles, Triangles, Squares, and harmonical Proportions; which engaged the Priefts and Philofophers to reprefent the divine and natural Operations by fuch Figures: in which they were followed by Pythagoras, Plato, &c.

But it must be obferved, that this Ufe of Geometry among the Ancients was not ftrictly scientifical, as among us; but rather fymbolical: they did not argue, or reduce Things and Properties unknown from Lines; but represented or delineated Things that were known. In Effect, they were not used as Means or Inftruments of discovering, but Images or Characters, to preserve, or communicate the Discoveries made.

D

G

DEFINITION S.

Fig. 1. Geom.
Plate 1..

A

Of a POINT.

Point is that which has no Parts; that is, it has no Length, Breadth, nor Thickness. But as no Operation can be performed without the Affiftance of vifible and corporeal things, we must therefore reprefent the mathematical Point by the natural one, which is an Object of our Sight, the fmalleft and leaft fenfible, and is made by the Prick of a Pen or Pencil, as the Point marked A.

A central Point, or Center, is a Point from whence a Circle or Circumference is defcribed; or rather, it is the Middle of a Figure, as the Point B.

A fecant Point, is a Point through which Lines cross each other, and is ufually called a Section. C.

Of