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THE

PREFACE

N the Preface to the laft Book, I gave an Account of the Ufefulness of Mathematicks in general : **** I fall now do it with Regard to Particulars. And,

Ift, Of ARITHMETICK, which is undoubtedly the most extensive and neceffary of all others: Without this, Trade and Commerce would foon be at an End, and the other Branches of the Mathematicks uncertain and defective.

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2dly, TRIGONOMETRY: By this Science we come to the Knowledge of a vast many Things of abfolute Importance. to Mankind: Hereby we understand the Motions, Diftances and Figures of Celeftial Bodies; by its Affifiance Geographers measure the Distances of Places fituate upon the Surface of the Earth; and on this Art does Navigation entirely depend, it being neceffary to determine the Pofition of the Ship, either with Regard to the Place failed from,or the Port intended for by aTrigonometrical Calculation: In a Word it is impoffible to live well without it; and the most useful Parts of Knowledge would be utterly loft, if Mankind were ignorant of it. PLATO was wont to Say, That ARITHMETICK and TRIGONOMETRY were the two Wings on which the Mind of Man mounted itself up to the Heavens; meaning thereby, their Ufefulness in determining the Motions, and difcovering the Harmony of the Celestial Bodies.

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3dly, GEOMETRY: By this Science we learn to compare Magnitudes and Meafures, and thereby discover the Ratio, or Habitude, one Thing bears unto another. There is no State of Life, which does not require the Use of Measures; all the Arts, which tend to the Supplying of Mankind with Food and Raiment, fubfift only by Meafures of Lengths, Breadths, Circles, Squares, &c. and the Capacities of dry and liquid Things, and their Reductions. Take arvay Geometry from any Art, either Liberal or Mechanic, and you take away its Soul or Reafon; that is, you deftroy it. 'Tis pretended, the Egyptians were the first Inventors of GEOMETRY, because they were the first that were under any Neceffity of preferving the Limits and Extent, of their Lands and Poffeffions, either by Figures traced and delineated, or by Accounts and Memorials contain'd in Writing, the Inundations of their Lands by the Overflowing of the NILE effacing all other Marks. Yet, (as an ingenious Author remarks) as there was the fame Neceffity for it before the Low Parts of EGYPT were inhabited, fince there was no making any Divifion, without Taking the Measures of the whole, and of each Part; nor any building of Houfes or Towns, without confidering their Figures, and regulating them by Measures: 'Tis to be preJumed, that the Science of Meafuring is as old as the World. Man took from himself the Measures of all other Things; he did not borrow from without the Names of Fathoms, Cubits, Feet, Handbreadths, Inches, &c. but he apply'd bis Thumb, bis Hand, his Feet, or his Arms, to the Things he would measure; and fo took from his own Body the Dimenfions of all other Things. VITRUVIUS fays, That the Græcians, through an Excess of Delicacy in Building, did not only take the Parts of Humane Bodies for the common Standard of their Plans and Draughts, but even the Beauty of Human Bodies for the Model of their Architecture: And hence they formed the Corinthian Order, from the beautiful Proportion of a well-shaped Woman; the Ionic, from the Shape of a handsome well proportioned Man; and the Doric, from the Shape of a labouring Man, ftrong and robuft. Man imagined himself the most perfect of created Beings, and fo propofed his own Body as the Model and Measure of all others.

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4tbly, ALGEBRA, which is of great Uje in helping us to diftinguifh the Relations of different Species of Quantities, and inveftigating of Theorems: It is a Method of Reafoning by the Help of Letters, representing Quantities, and is to the Mathematicks, the fame that Logic is to the ordinary Philofophy, and is therefore fometimes called Logistic.

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5thly, The Analyfis of INFINITES, or FLUXIONS, which is only a different Method of using the Characters of Common ALGEBRA; by which admirable Invention of the Great Sir Ifaac Newton, we are much affifted in every Part of the Mathematicks. This is One of the most fublime Difcoveries, the World was ever yet acquainted "Tis a with, and perhaps we may expect no further. pleafant Thing (fays Mr. DITTON) to contemplate the Jeveral Gradations and Steps made in the Improvement of this Method: The old Geometers used the Method of "Exhauftions to admirable good Purpose; but this, after "a long Tract of Time, gave Place to a fuperior one; namely, that of Indivifibles. And in much less Time, was this improved and heightned into the noble Arith"metick of Infinites. And (Geometry being now in hafty "Growth towards its Maturity) in a very short Time the "Method of Fluxions appeared.

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"But what a wide Step is it from the Method of Ex"bauftions hither? How humble and low was the glo "rious Structure, when the first ARCHIMEDES left it, "in Comparison to what it is, now the SECOND has fet "his finishing Hand to it? If there are none, who confi"der thefe Things without Wonder and Delight, there are

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fome who go farther, and are ready to ask the Question, "Whether the next Step will not be to the Algebra of "Pure Intelligences? However that be; 'tis certain, this "Method helps a Man to arrive at Truth, by a most easy, "cheap, and compendious Way. And it is for that very "Reafon, by fo much more perfect than all others, by how "much the fewer (and thofe more facile) Steps, it takes to "obtain its End. All the Works of Nature have a charming Simplicity and Concifeness in them: And there is fo

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" much

"much of that Beauty bere, that this alone were fufficient to recommend it for the genuine Offspring of Eternal "Truth and Reafon.

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MAY the 25th, 1734.

Advertisement to all Lovers of Ingenuity.

THE Author intending to publish the next Quarterly Book in September enfuing, defires that fuch Gentle men and Ladies, &c. as think fit to encourage this Undertaking, either by promoting and recommending the Sale thereof, or by fending fuch Subjects, either in Verfe or Profe, as are fuitable to this Defign, would pleafe to direct their Letters to Mr. THO. GENT, Printer, near the Star in Stone-Gate, YORK, Post paid, before the End of Auguft next, where they will be carefully forwarded to the Author.

Whoever fend either Enigmas, or Arithmetical Questions, are defired to give the Solution along with them, to prevent all Miftakes, or Ambiguity in the Terms: And I do affure the Buyers of this Treatife, that, in Procefs of Time, they fball have Questions in all the Branches of Mathematicks, viz. Arithmetical, Nautical, Geographical, Mechanical, Trigonometrical, Aftronomical, Algebraical, and in Fluxions, together with exponential Equations, and the Con ftruction of Algebraic Equations, and finding their Loci; to fuit the Taftes of all Perfons, and render the Work ge nerally useful.

Mifcellanea

C

Mifcellanea Curiofa, &c.

ACONTINUATION of Dr. R. COTES'S PREFACE.

I

F the Bodies of the Planets and Comets are carried round the Sun in Vortices; the Bodies fo carried, and the Parts of the Vortices next furrounding them, muft be carried with the fame Velocity and the fame Direction, and have the fame Denfity, and the fame Vis Inertie, anfwering to the Bulk of the Matter. But it is certain, the Planets and Comets, when in the very fame Parts of the Heavens, are carried with various Velocities, and various Directions. Therefore it neceffarily follows, that those Parts of the Cæleftial Fluid, which are at the fame Diftances from the Sun, muft revolve at the fame time with different Velocities, in different Directions: For one kind of Velocity and Direction is required for the Motion of the Planets, and another for that of the Comets. But fince they cannot be accounted for, we muft either fay, that all the Cæleftial Bodies are not carried about by Vortices, or elfe that their Motions are derived, not from one and the fame Vortex, but from diftinct ones, which fill and pervade the Spaces round about the SUN.

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BUT if feveral Vortices are contained in the fame Space, and are fuppofed to penetrate each other, and to revolve with different Motions; then because these Motions must agree with thofe of the Bodies carried about by them, which are perfectly regular, and performed in

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