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THE Human Particulars are ten.

1. The Stature of the Inhabitants; their Meat, Drink, and Origin.

2. Their Arts, Profits,

Trade.

Commodities, and

3. Their Virtues and Vices; their Capacities and Learning.

4. Their Ceremonies at Births, Marriages, and Funerals.

5. Their Speech and Language.

6. Their Political Government.

7. Their Religion and Church Government. 8. Their Cities.

9. Their memorable Hiftories.

10. Their famous Men and Women, Artificers, and Inventions.

CHA P. II.

Some Propofitions in Geometry and Trigonometry, of ufe in Geography.

PH

ATO very justly called Geometry and Arithmetic the two Wings whereby the Minds of Men might mount up to Heaven; that is, in fearching after the Motions and Properties of the Celestial Bodies. Thefe Sciences are no less useful in Geogra phy; if we defire to understand it's fublime and intricate Parts, without any Hinderance. It is true, a less share of Mathematics will ferve for Geography, than Aftronomy: but because several are taken with

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the Study of Geography, who do not understand thefe Sciences, we shall here fet down a few Propofitions from them, fuch as we think moft neceffary; that the Reader may proceed the more readily without Interruption in his Study. Tho', by the way, we do not at all encourage that bad Custom fome young Gentlemen have got, in applying themselves unadvisedly to other Parts of Philofophy, before they have a competent, Knowledge in Arithmetic and Geometry. The Fault is very often in their Masters and Tutors, who are for the most Part ignorant of thefe Things themfelves, and therefore cannot admonish Youth to fhun fo pernicious a Cuftom. In Arithmetic we fuppofe the Reader to know the four common Rules of Numeration, viz. Addition, Subftraction, Multiplication, and Divifion, with the Golden Rule, or Rule of Three; and therefore shall not treat of them here. If any one understand them not, he may learn them much better from fome able Teacher, than from Books.

1. BUT as to Geometry, it treats of three forts of Magnitudes by which every thing is meafured; viz Lines, Superficies, and Solids: neither can there be found in Nature a Body of any other Dimenfion.

2. A LINE is either ftraight or curved; and a Curve again is either uniform as circular, or diffimilar and variable, as the Ellipfe, the Conchoid, and Spiral Line.

3. A CIRCLE is a Space or plain Superficies bounded with a curve Line, wherein there is a Point from which all right Lines drawn to the Curve are equal. The curve Line which bounds that Space is called the Circumference, or Periphery of the Circle; and the middle Point is called the Center (a).

(a) Euclid Lib. 1. Def. 15, 16.

4. THE Diameter of a Circle is a right Line drawn thro' the Center, and terminated at both ends by the Periphery: one half of which is called the Semidiameter, or Radius (a).

5. AN Arch is part of the Periphery of a Circle. A Quadrant is a fourth Part of the whole Periphery. What an Arch wants of a Quadrant is called the Complement of that Arch and it's Difference from a Semicircle is called it's Supplement (b).

PROBLEM.

6. HAVING a right Line given and a Point either in, or out of it, to draw thro' that Point a Line perpendicular to the former.

LET the Line given (Fig. 2.) be A B, and the Point C: open, the Compaffes fo, that fetting one Foot in C, you may with the other cut the Line given in df; then one Foot being placed at d, with the other defcribe an Arch, as gb; alfo make ƒ the Center, and with the fame Radius defcribe another Arch, which will cut the former in g and b; fo draw the Line gb; which will be the Perpendicular required.

7. TO divide a Circle and it's Periphery into four equal Parts. Draw a Diameter, and from the Center raise to it a Perpendicular, which prolong'd will be alfo a Diameter; whereby both the Circle and it's Periphery will be divided into four equal Parts (c).

8. TO divide the Periphery of a Circle into Degrees. A Degree is the 360th Part of the Circumference. Mathematicians always divide the Periphery into

(a) Euclid Lib. 1. Def. 17.
(b) lb. Prop. 11, 12. Lib. 1.
VOL. I.

(c) Ibid. Prop. 4. Lib. iv.

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fo many equal Parts (d); and each of these Parts into 60 fmaller Divifions, called firft Minutes; alfo each Minute into 60 Seconds, &c. commonly writ thus, 3 degr. 2. min. 5 fec. that is, 3 Degrees, 2 Minutes, 5 Seconds. Hence the Quadrant containeth 90 Degr. the Semicircle 180, and the fixth Part of a Circle 60 Degrees.

THEREFORE to folve this Problem, divide the Periphery into Quadrants, then take off the Semidiameter, and with it's Length cut an Arch from the Periphery (e), which will be equal to 60 Degr. fo there remains in the fame Quadrant 30 Degr. which being bifected you will have 15 Degr. this again mechanically trifected will give 5 Degr. which divided into five equal Parts make fo many Degrees, Q. E. F. But this is done more artificially by mathematical Inftruments (ƒ).

9. To find the Area of a Quadrangle, or a Space contained in a Figure of four Sides, and four Right Angles. Multiply one fide by the other, and the Product is the Area. It is to be obferved that Lines are measured by Lines, and Superficies, by Measures that are Superficies, or Squares; alfo the Contents of folid Bodies, which have their Dimenfions, are computed in folid Measure, or fo many Cubes. Thus we measure the Sides of a House by a lineal Foot, the Floors and Wainscot by a

(d) This Divifion of a Circle into 360 Parts, or Degrees, is becaufe that number can be divided into more Aliquot Parts, than any other convenient Number, viz. into 2, 3, 4, 5, 6, 8 and 9 Parts.

(e) Euclid. Prop. 15. Lib. iv. (f) By a Line of Chords truly divided; thus, from any Point in the Periphery lay on the Chord of one Degr. then

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from the fame Point lay on the Chord of two Degr. so of three Degr. &c. 'till you come to 90 Degr. then begin again as before, 'till the whole Periphery is divided. By this means you will avoid the Errors which may arife from the intermediate Divifions; and tho' thefe Errors fingly confidered are very small, yet in fo many Degr. they will produce one very fenfible.

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square Foot, and the Space it encloseth, confider ed as a Solid, by a cubical Foot.

10. HAVING the Diameter or Semidiameter of a Circle, to find the Periphery in the fame Measure: and converfly, having the Periphery given, to find the Diameter as near as poffible (g). The Solution of this Problem depends upon the determined Proportion of the Diameter to the Periphery, which is nearly as 7 to 22; as is demonftrated by Archimedes; or more accurately, as 10000000000 is to 31415926535 (b). For Example, let the Diameter be 12 Foot; by the Golden Rule, as 7 is to 22 fo is 12 to the Periphery of the Circle; or if you use the other Proportion it will be much the fame.

BUT if the Periphery be given, and the Diameter be required, fay; as 22 is to 7, or as 31415926535 to 10000000000, fo is the Periphery given to the Diameter required.

II. THE Diameter and Periphery of a Circle, or either of them, being given in Miles or Feet, to find

(g) See Tacquet's felect The orems of Archimedes, Prop. 5. (b) Tho' it be well known that the Periphery of a Circle is incommenfurable to the Diameter, yet either of these Proportions will ferve well enough for common Ufe. But no Proportion in fmall Numbers is fo exact as that of Andrew Metius, viz. of 113 to 355, which is found not to differ from the Truth above. But if the Reader defireth the niceft Computation of the Proportion of the Diameter of a Circle to, the Circumference (altho' that of Matius comes very near), let him have recourse to the laborious Calculus of Van Ceulen,

who carried his Calculation to
35 places of Decimal Fractions.
Or if he would still be more
nice and curious, he may have
recourfe to Mr Abr. Sharp's
Calculation, to double the Nam-
ber of Vau Ceulen's Fractions.
By which Exactness, the Cir-
cumference of the Terraqueous
Globe, may be computed to a
a Degree less than the Breadth
of a Grain of Sand: yea, more
than this, the number of the
Grains of Sand, that would be
contained in a Space as big as
the Sphere of the Fixt Stars,
might be truly computed by
this means. Vid. Math. Tables
printed for Mr Mount, page
53, &c.
C 2

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