the Area of that Circle in fquare Miles, or fquare Feet. Multiply one half of the Periphery into the Semidiameter, and the Product will be the Area required (i) but if you have only one of them given, you may find the other by the last Problem: Or it may be done without it (k).

12. THE Diameter, or Semidiameter, of a Globe being given; to find it's Superficies in Square, or it's Solidity in Cubic Measure.

A Globe is a round folid Body, having a certain Point in the Center of it, from whence all right Lines drawn to the Surface are equal: and a Line drawn thro' this Point is the Diameter, about which if the Globe be revolved it is called it's Axis (1). Moreover if a Globe be cut any how by a right Line, the Section is a Circle; if thro' the Center the Circle will have the fame Diameter as the Globe itfelf; and fuch are called the greater Circles of the Sphere or Globe, and the reft leffer Circles. To folve the Problem (m): By the tenth Article, find the Periphery; then multiply the Diameter into this Periphery, and the Product will be the Superficies of the Globe in fquare Measure, which multiplied into the % of the Diameter, will produce the Solidity of the Globe in cubic Measure.

(m) See this demonftrated in Tacquet's Select Theorems of Archimedes, Scholium 2 of Prop. 24. and that of Prop. 28.

13. A RIGHT angled Triangle bath two fides perpendicular to each other (or make an Angle of 90 Degr.) which two fides are called the Catheti, or Perpendiculars, and the third fide the Hypotenufe.

THE Measure of an Angle is the Length of an Arch described from the angular Point as a Center: that is, as many Degrees as the Arch between the Legs of the Angle doth contain; fo many Degrees the Angle is faid to be of. Thus a right Angle is 90 Degr. because the Arch fo defcribed is a Quadrant.

THE right Sine of an Arch is a right Line drawn from the one end of the Arch perpendicular to the Diameter, which paffeth thro' the other end (n).

THE Tangent of an Arch is a right Line which touches the Arch at one end, and is bounded at the other with a Line drawn thro' the Center, and the other end of the Arch; which Line is called the Secant of that Arch.

MOREOVER, the Sine, Tangent, and Secant, of an Angle, are the fame of the Arch which measureth the Angle.

(n) Mr Whiston in his Notes upon Tacquer's Euclid, has neatly explained the Origin of Sines, Tangents, and Secants. Coroll. to the 47th Prop. Lib. i. which we shall here tranfcribe. Let AC the Semidiameter of a Circle (Fig. 3) be of 100.000 Parts, and the Angle BAD of 30 Deg. because the Chord or Subtenfe of 60 Degr. is equal to AC the Semidiameter (by Prop. 15. Lib. iv. Euclid) BD the Sine of 30 Degrees fhall be equal to one balf the Semidiameter, or AC; and therefore shall contain 50.000 Parts. But now in the right-angled Triangle

ABD, the Square of AB is equal to the Square of AD and BD. Therefore let the Semidiameter AB be fquared, and from that Square fubftract the Square of BD: The Remainder will be the Square of AD or of the Co-fine B F equal to it: out of which extract the fquare Root, and you will have the Line BF or AD. Then by this Analogy as A B: BD::AE: CE or AD: BD::AC: CE, fo you have the Tangent CE. And if the Square of AC be added to the Square of CE, the Root of the Sum being extracted will be the Secant AË. Q. E. I.

IT is alfo neceffary to be known that Tables have been calculated by the great Labour and Industry of fome Mathematicians, in which the Diameter being taken for 100000, &c. the Sines, Tangents, and Secants, are found out in proportional Numbers; as of 2 Degr. 10 Degr, 20 Degr. 32 Min. &c. Thefe Tables are called mathematical Canons, and are of extraordinary ufe in all mathematical and phyfical Sciences; wherefore I am willing to give fome Hints of these things to the young Geographer. But because spherical Triangles have fome Difficulty in their management, and regard none but thofe who defire to be deeper fkilled in this Science, we fhall pafs them by; and only treat of right-angled Triangles, the measuring of which is as eafy as neceffary.

Two THEOREMS,

14. THE three Angles of every Triangle, taken together, are equal to two right Angles, or 180 Degr and therefore the two acute Angles of a right angled Triangle make exactly 90 Degr. (o). Also if a right Line touch a Circle, and there be drawn from the Point of Contact another right Line to the Center, that Line makes a right Angle with the Tangent (p).

15. THE most neceffary Problems are these.

I. THE Hypotenuse and one fide of a right angled Triangle being given, to find either of the acute Angles. Say by the Golden Rule; As the given Hypotenufe is to the given fide: fo is the Radius 100000 (which Number is affumed equal to the Semidiameter in the Tables) to the Sine of the oppofite Angle; which Sine being found in the Tables

(0) Euclid, Prop. 33. Lib. i.

(p) Ibid. Prop, 18. Lib. iii. will

will fhew the Quantity of the Arch or Angle oppofite to the Side given; and the other Angle is the Complement of that now found, to 90 Degr.

II. ONE fide and the acute Angle next it being given, to find the Hypotenufe. Say as before; As the Sine of the Complement of the given Angle is to the Radius 1000000: fo is the Side given to the Hypotenuse fought.

III. HAVING two Sides given, to find either of the acute Angles. Say, As either of the Sides is to the other, fo is the Radius 100000 to the Tangent of the Angle adjacent to the Side first affumed.

IV. HAVING the Hypotenuse and one acute Angle given, to find either of the Sides: Say; As the Radius 100000 is to the Sine of the Angle oppofite to the Side required: So is the given Hypotenuse to that Side.

Of Divers Measures.

BECAUSE the ufe of Measures is frequent in Geography, and fince divers Nations ufe different Measures, 'tis proper to premise fomewhat concerning them; partly that the Reader may the better understand the Writings of the antient Geographers and Hiftorians; and partly that he may compare together those in ufe at this Day.

THE Length of a Foot is almost universally made use of, tho' a Foot in one Place differs from that in another. Mathematicians frequently measure by the Rhinland Foot of Snellius, which he proves to be equal to the old Roman Foot. And because Snellius was very diligent and accurate in measuring the Earth, that Rhinland Foot

of his is defervedly taken as a Standard for all other Measures (q). See half it's Length, Fig. 1.

A PERCH or Pole ought to confift of ten fuch Feet. But the Surveyors in Holland make 12 Feet a Rhinland Perch, and in Germany they compute 16; which is very incommodious in Calculation. Snellius makes the Holland Mile equal to 1500 Rbinland Perches (each Perch being 12 Foot) or 1800 Rhinland Feet.

THESE two Measures, a Perch and a Mile, arife from the repetition of a Foot; but a Palm, an Inch, and a Barley-Corn (which are fometimes ufed in Holland) proceed from it's Divifion. An Inch is the twelfth Part of a Foot. A Palm is 4 Inches. A Barley-Corn is the fourth Part of an Inch. However it would be much better to divide the Foot into 10 Inches, and the Inch into 10 Subdivifions or Seconds, &c.

THESE are the Meafures now made use of by the Dutch in Geography. It remains that fome others be alfo taken Notice of; viz. thofe of the antients, whether Greeks, Romans, Perfians, Ægyptians; and thofe alfo of later Times as of the Turks, Polanders, Germans, Mofcovites, Italians, Spaniards, French, and English.