the Area of that Circle in square Miles, or square 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 Semidiameler, of a Globe being given; to find it's Superficies in Square, or it's Solidity in Cubic Measure. A Globe is a round solid 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(/). 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 itself; and such are called the greater Circles of the Sphere or Globe, and the rest lesser Circles. To solve the Problem (/»): 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 square Measure, which multiplied into the i of the Diameter, will produce the Solidity of the Globe in cubic Measure. (/) As is demonstrated by j I I I Archimedes, Prop. I. De Di- J- — — -j- — —■ menfione Circuit. 3 5 7 9 (if) By saying, as the Square i I I I of i (which is i) is to -7854 (the Area of a Circle whose Diameter is 1) so is the Square I I of any other Diameter to it's — — -j— —, &•<;. Area; By Prop. z. Lib. ii. of 19 1 21 Euclid. The famous M. Leib- (I) Euclid. Lib. ii. De/. 14, nitz has demonstrated that if Ij, 16, 17. the Diameter of a Circle be I, (m) See this demonstrated in the true Area will be ± — ^ Theorems of Ar I chtmedes, Scholium 2 of Prop. 24. and that of Prop. 28. • 13. A RIGHT angled Triangle hath two sides perpendicular to each other (or make an Angle of 90 Degr.) which two fides are called the Catheti, or Perpendiculars, and the third side the Hypotenuse. 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 so many Degrees the Angle is said to be of. Thus a right Angle is 90 Degr. because the Arch so described 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 pafleth thro' the other end (»). 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. M O R E O V E R, the Sine, Tangent, and Secant, of an Angle, are the fame of the Arch which measureth the Angle. (») Mr Whiston in his Notes A B D, the Square of AB is e upon Tacquefs Euclid, has neat- qua/to the Square of AD and ly explained the Origin of Sines, BD. Therefore let the Semi Tangents, and Secants. Coroll. diameter AB be squared, and to the 47th Prop. Lib., i. which from that Square subfiract the we shall here transcribe. Let Square «/BD: The Remainder AC the Semidiameter of a Circle will be the Square of AT) or (Fig- $ be of 100.000 Parts, of the Co-fine B F equal to it: and the Angles AD of 30 Deg. out of which extract the square because the Chord or Subtense of Root, and you will have the 60 Degr. is equal to AC the Line BF or AD. Then by this Semidiameter {by Prop. I 5. Lib. Analogy as A B : B D : : A E: iv. Euclid) BD the Sine of 30 CE or AD : BD : : AC : CE, Degrees Jhall be equal to one so you have the Tangent CE. half the Semidia meter, or 5 And if the Square of'A C be ad A C; and therefore Jhall con- ded to the Square of CE, the tain 50.000 Parts. But now Root of the Sum being extracted in the right-angled Triangle will be the Secant AE. Q-E. I. I T is also necessary to be known that Tables have been calculated by the great Labour and Industry of some Mathematicians, in which the Diameter being taken for iooooo, &c. the Sines, Tangents, and Secants, are found out in proportional Numbers •, as of 2 Degr. 10 Degr. 20 Degr. 3 2 Min. tsV. These Tables are called mathemartical Canons, and are of extraordinary use in all madiematical and physical Sciences; wherefore I am willing to give some Hints of these things to the young Geographer. But because spherical Triangles have some Difficulty in their manager ment, and regard none but those who desire to be deeper skilled in this Science, we shall pass them by i and only treat of right-angled Triangles, the measuring of which is as easy as necessary. Two THEOREMS, 14. THE three /ingles 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. (0). Also if a right Line touch a Circle, and there be drawn from the Point of Contacl another right Line to the Center? that Line makes a right Angle with the Tangent (p). 15. THE most necessary Problems are these. I. THE Hypotenuse and one side of a right angled Triangle being given, to find either of the acute Angles. Say by the Golden Rule; As the given Hypotenuse is to the given side: so is the Radius iooooo (which Number is assumed equal to the Semidiameter in the Tables) to the Sine of the opposite Angle; which Sine L being found in the Tables (<0 Euclid. Prop. 3?. Lit. i. (/) Ibid. Prop, 18. Lib. Hi. will will mew the Quantity of the Arch or Angle opposite 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 Hypotenuse. Say as before •, As the Sine of the Complement of the given Angle is to the Radius 1000000: so is the Side given to the Hypotenuse sought. III. HA VING two Sides given, to find either of the acute Angles. Say, As either of the Sides is to the other, so is the Radius 100000 to the Tangent of the Angle adjacent to the Side first: assumed. 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 opposite to the Side required: So is the given Hypotenuse to that Side. .; Of Divers Measures. BECAUSE the use of Measures is frequent in Geography, and since divers Nations use dffferent Measures, 'tis proper to premise somewhat concerning them; partly that the Reader may the better understand the Writings of the antient Geor graphers and Historians •, and partly that he may compare together those in use 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 SneUius, which he proves to be equal to the old Roman Foot*. And because Snellius was very diligent and accu* rate in measuring the Earth, that Rhinland Foot of his is deservedly taken as a Standard for all other Measures (q). See half it's Length, Fig. i. A P E RC H or Pole ought to consist of ten such Feet. But the Surveyors in Holland make 12 Feet a Rbinland 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 Rbinland Feet. THESE two Measures, a Perch and a Mile, arise from the repetition of a Foot; but a Palm, an Inch, and a Barley-Corn (which are sometimes used in Holland) proceed from it's Division. 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 1 o Inches, and the Inch into 10 Subdivisions or Seconds, £sV. THESE are the Measures now made use of by the Dutch in Geography. It remains that some others be also taken Notice of; viz. those of the antients, whether Greeks, Romans, Persians, Ægyptians; and those also of later Times as of the Turks, Polanders, Germans, Moscovites, Italians, Spaniards, French, and Englijh. |