has to any other place, as to the diftance and bearing, its Climate, Heat, Cold, length of days, &c. which things do much enliven the Reader's notion of the very action itself. Juft fo, it is neceflary to know the Technical or Doctrinal part of Chronology, if a Man would be thoroughly skill'd in Hiftory, it being impoffible without it, to unravel the confufion of Hiftorians. I remember Mr. Hally has determin'd the day and hour of Julius Cæfar's Landing in Britain, from the circumftances of his relation. And every body knows, how great use our incomparable Hiftorians Mr. Dodwell has made of the Calculated times of Eclipfes, for fettling the times of great Events, which before were as to this effential circumftance almoft fabulous. Both Chronology and Geography, and also the knowledge of the Sun's and Moon's motions, fo far as they relate to the conftitution of the Kalendar and Year, are neceffary to a Divine, and how fadly fome otherwise Eminent have blunder'd, when they meddled with things that relate to these, and border on them, is too apparent.

Per

No body, I think, will queftion the intereft, that Mathematicks have in Painting, Mufick, and Architecture which are all founded on Numbers. fpective and the Rules of Light and Shadows are owing to Geometry and Opticks: And I think thofe two comprehend pretty near the whole Art of Painting, except decorum and ordinance; which are only a due obfervance of the History and Circumstances of the fubject, you reprefent. For by Perfpective, may be understood the Art of defigning the outlines of your folid, whether that be a Building, Landfkip, or Animal: and the draught of a Man is really as much the Perfpective of a Man, as the draught of a Building is of a Building; tho' for particular reafons, as because it confifts of more crooked lines,

B 2

&c. it is hard to reduce the Perspective of the former, to the ordinary established Rules.

If Mathematicks had not reduced Mufick to a regular Syftem, by contriving its Scales, it had been no Art, but Enthufiaftick Rapture, left to the roving fancy of every Practitioner. This appears by the extraordinary pains, which the Ancients have taken to fit numbers to three forts of Mufick, the Diatonick, Chromatick, and Enharmonick: which if we confider with their nicety in distinguishing their feveral Modes, we shall be apt to judge, they had fomething very fine in their Mufick, at leaft for moving the paffions with fingle Inftruments and Voices. But Mufick had been imperfect ftill, had not Arithmetick ftepped in once more, and Guido Aretinus by inventing the temperament making the Fifth falfe by a certain determined quantity, taught us to Tune our Organs, and intermix all the three kinds of the Ancients, to which we owe all the Regular and Noble Harmony of our modern Mufick.

As for Civil Architecture (of Military I shall speak afterwards) there is hardly any part of Mathematicks, but is fome way fubfervient to it. Geometry and Arithmetick for the Due measure of the feveral parts of a Building, the Plans, Models, computation of Materials, time and charges: for ordering right its Arches and Vaults, that they may be both firm and beautiful: Mechanicks for its ftrength and firmness, tranfporting and raifing materials: and Opticks for the Symmetry and Beauty. And I would not have any affume the character of an Architect without a competent fkill in all of thefe. You fee that Vitruvius requires these and many more for making a compleat Architect. I must own, that should any one set up to practice in any of the fore-mentioned Arts, furnished only with his Mathematical Rules, he would produce but very clumfy pieces. He,

that

that should pretend to draw by the Geometrical Rules of Perspective, or Compofe Mufick meerly by his skill in Harmonical numbers, would fhew but aukward performances. In those Compos'd Subjects, befides the ftiff Rules, there must be Fancy, Genius, and Habit. Yet nevertheless these Arts owe their being to Mathematicks, as laying the foundation of their Theory, and affording them Precepts, which being once invented, are fecurely rely'd upon by Practitioners. Thus many defign, that know not a tittle of the reafon of the Rules, they practice by; and many no better qualify'd in their way Compose Musick, better perhaps than he could have done, that invented the Scale, and the Numbers upon which their Harmony is founded. As Mathematicks laid the foundation of these Arts, fo they muft improve them and he, that would invent, must be skill'd in Numbers, befides it is fit a Man fhould know the true grounds and reafons of what he studies: and he that does fo, will certainly practice in his Art with greater judgment and variety, where the ordinary Rules fail him.

I proceed now to fhew the more immediate ufefulness of Mathematicks in Civil Affairs. To begin with Arithmetick, it were an endless task to relate its feveral uses in publick and private business. The regulation and quick dispatch of both, seem entirely owing to it. The Nations, that want it, are altogether barbarous, as fome Americans, who can hardly reckon above twenty. And I believe it would go near to ruin the Trade of the Nation, were the eafy practice of Arithmetick abolished: for example, were the Merchants and Tradesmen oblig'd to make ufe of no other than the Roman way of notation by Letters, inftead of our prefent. And if we fhould feel the want of our Arithmetick in the easiest Calculations, how much more in those, that are some

thing harder; as Intereft fimple and compound Annuities, &c. in which, it is incredible, how much the ordinary Rules and Tables influence the dispatch of bufinefs. Arithmetick is not only the great Inftrument of private Commerce, but by it are (or ought to be) kept the publick Accounts of a Nation: I mean thofe, that regard the whole State of a Common-wealth, as to the number, fructification of its people, increase of Stock, improvement of Lands and Manufactures, Ballance of Trade, Publick Revenues, Coynage, Military power by Sea and Land, &c. Thofe, that would judge or reafon truly about the State of any Nation, muft go-that way to work, fubjecting all the fore-mention'd particulars to Calculation. This is the true Political knowledge. In this refpect the affairs of a Common-wealth differ from thofe of a private Family, only in the greatness and multitude of particulars, that make up the accounts. Machiavel goes this way to work in his account of different Eftates. What Sir William Petty and feveral others of our Countrymen have wrote in Political Arithmetick, does abundantly fhew the Pleafure and ufefulnefs of fuch Speculations. It is true, for want of good informa tion, their Calculations fome times proceed upon erroneous fuppofitions: but that is not the fault of the Art. But what is it, the Government could not perform in this way, who have the command of all publick Records?

Laftly, Numbers are applicable even to fuch things, as feem to be govern'd by no rule, I mean fuch as depend on Chance: The quantity of probability and proportion of it in any two propofed cafes being fubject to Calculation as much as any thing elfe. Upon this depend the principles of Game. We find Sharpers know enough of this, to cheat fome men that would take it very ill to be thought

Bubs

Bubbles: And one Gamefter exceeds another, as he has a greater fagacity and readinefs in Calculating his probability to win or lofe in any propofed cafe. To understand the Theory of Chance thoroughly, requires a great knowledge of Numbers, and a pretty competent one of Algebra.

The feveral uses of Geometry are not much fewer than those of Arithmetick. It is neceffary for ascertaining of property both in Plains and Solids, or in Surveying and Gauging. By it Land is fold by the meafure as well as Cloth: Work-men are pay'd the due price of their labour, according to fuperficial or folid measure of their work: and the quantity of liquors determined for a due regulation of their price and duty. All which do wonderfully conduce to the easy dispatch of business, and the preventing of frauds and controverfies. I need not mention the Measuring distances, laying down of Plans, and Maps of Countries, in which we have daily Experience of its usefulness. These are fome familiar inftances of things, to which Geometry is ordinarily apply'd of its ufe in Civil, Military, and Naval Architecture we shall speak afterwards.

From Aftronomy we have the regular difpofition of our time, in a due fucceffion of years, which are kept within their limits as to the return of the Seafons, and the motion of the Sun. This is no small advantage for the due repetition of the fame work, Labour and Actions. For many of our Publick, Private, Military, and the Country Affairs, Appointments, &c. depending on the products of the Ground, and they on the Seafons; It is neceffary, that the returns of them be adjusted pretty near to the motion of the Sun: and we should quickly find the inconveniency of a vague undetermined year, if we used that of the Mahumetans, whofe beginning and every month wanders through all the days of ours B 41

or