THE Human Particulars are ten. "i. The Stature of the Inhabitants; their Meat, Drink, and Origin. 2. Their Arts, Profits, Commodities, and Trade. 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 Histories. 10. Their famous Men and Women, Artificers, and Inventions. Some Propositions in Geometry and'Trigonometry, of use in Geography. PLATO very justly called Geometry and Arithmetic the two Wings whereby the Minds of Men might mount up to Heaven; that is, in searching after the Motions and Properties of the Celestial Bodies. These Sciences are no less useful in Geography; if we desire to understand it's sublime and intricate Parts, without any Hinderance. It is true, a less share of Mathematics will serve for Geography, than AJlronomy: but because several are taken with the the Study of Geography, who do not understand these Sciences, we shall here set down a few Propositions from them, such as we think most necessary j 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 some young Gentlemen have got, in applying themselves unadvisedly to other Parts of Philosophy, 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 these Things themselves, and therefore cannot admonilh Youth to shun so pernicious a Custom. In Arithmetic we suppose the Reader to know the four common Rules of Numeration, viz. Addition, Subjlrailidn, Multiplication, and Division, 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 some able Teacher, than from Books. 1. BUT as to Geometry, it treats of three sorts of Magnitudes by which every thing is measured; viz Lines, Superficies, and Solids: neither can there be found in" Nature a Body of any other Dimensions. 2. A LINE is either straight or curved ; and a Curve again is either uniform as circular", or dissimilar and variable, as the Ellipse, the Conchoid, and Spiral Line. 3. A CIRC L E 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). la) Euclid Lib. i. Des. 15, 16. 4. THE 4. T H E 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 (Pig; 2..): be AB, and the Point C: open.the Compasses so, that setting 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 describe.an Arch, asg h also make / the Center, and with the fame Radius describe another Arch, which will cut the former in g and h; so draw the Line g b; 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 also 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. i. Def. 17. [c) Ibid. Prop. 4. Lib. iv. . (b) Ib. Prop. 11, 12. Lib. 1. VOL. I. C so so many equal Parts (d); and each of these Parts into 60 smaller Divisions, called first Minutes; also each Minute into 60 Seconds, &c. commonly writ thus, 3 degr. 2. min. 5 sec. that is, 3 Degrees, 2 Minutes, 5 Seconds. Hence the Quadrant contained! 90 Degr. the Semicircle 180, and the sixth Part of a Circle 60 Degrees. THEREFORE to solve this Problem, divide the Periphery into Quadrants, then take off the Semidiameter, and with it's Length cut an Arch from the Periphery (<?), which will be equal to 60 Degr. so there remains in the fame Quadrant 30 Degr. which being bisected you will have 15 Degr. this again mechanically trisected will give 5 Degr. which divided into five equal Parts make ib many Degrees, Q^ E. F. But this is done more artificially by mathematical Instruments (/). 9. TO find the Area os a Quadrangle, or a Space contained in a Figure os sour Sides, and four Right Angles. Multiply one side by the other, and the Product is the Area. It is to be observed that Lines are measured by Lines, and Superficies, by Measures that are Superficies, or Squares; also the Contents of solid Bodies, which have their Dimensions, are computed in solid Measure, or so many Cubes. Thus we measure the Sides of a House by a lineal Foot, the Floors and Wainscot by a (//) This Division of a Circle from the fame Point lay on the into 560 Parts, or Degrees, is Chord of two Degr. so of three because that number can be di- Degr. &c. 'till you come to 90 vidcd into more Aliquot Parts, Degr. then begin again as before, than any other convenient 'till the whole Periphery is di Number, viz. into 2, 3, 4, 5, vided. By this means you will 6, 8 and q Parts. avoid the Errors which may a (n) Euclid. Prop. 15. Lib. iv. rife from the intermediate Di (/) Dy a Line of Chords visions; and tho' these Errors truly divided; thus, from any singly considered are very small, Point in the Periphery lay on yet in so many Degr. they will the Chord of one Degr. then produce one very sensible. 2 square square Foot, and the Space it encloseth, consider* 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 converse having the Periphery given, to find the Diameter as near as possible (g). The Solution of this Problem depends upon the determined Proportion of the Diameter to the Periphery, which ig nearly as 7 to 22 •, as is demonstrated by Archimedes; or more accurately, as ioocooeoooo is to 31415926535 (h). For Example, let the Diameter be 12 Foot; by the Golden Rule, as 7 is to 22: so is 12 to the Periphery of the Circle; or if you use the other Proportion it will be much the same. B U T if the Periphery be given, and the Diameter be required, say; as 22 is to 7, or as 31415926535 to ioooooooooOj so is the Periphery given to the Diameter required. 11. THE Diameter and Periphery of a Circle, or either of them, being given in MileS or Feet, to find (g) See Racquet's select Theorems of Archimedes, Prop. 5. [h) Tho' it be well known that the Periphery of a Circle is incommensurable to the Diameter, yet either of these Proportions will serve well enough for common Use. But no Proportion in small Numbers is so exact as that of Andrew Metius, viz. of 113 to 355, which is found not to differ from the Truth above To0d3o,oo. But if the Reader delireth the nicest 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, C 2 the who carried his Calculation to 3 5 places of Decimal Fractions. Or if he would still be more) nice and curious, he may have recourse to Mr Abr. Sbarp'i Calculation, to double theNumber of Vdu Ceitkn'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 S3> |