T The GOLDEN RULE, O work the Golden Rule, or Rule of Three, Geometrically, is, by having of Three Lines (or Numbers) given, to find a Fourth Line (or Number) that shall be in Proportion to them; that is, As the First Line (or Number) Is to the Second Line (or Number), Therefore, Having Three Lines (or Numbers) given, let it be required to find a Fourth that shall be in Proportion to them. L ET the Three Lines given be A 24, B 28, and C 36, of any Meafure), and let it be required to find a Fourth Line D, which fhall be in Proportion to them. First, Draw two lines, as E M and EF, making any Angle, as the Angle DEF; then take the line A in your Compaffes, and fet it from E to G ; then take the line B, and fet it from E to H, and draw the line GH; then take the third line C in your Compaffes, and fet that from E to K, upon the line E M, the fame where the firft line A was fet; and through the point K draw the line K L Parallel to G H, cutting the line EF in the point L; fofhall the line E L be the Fourth Proportional Line required. For, For, As the Line A, equal to E G 24, Yards of And fo, any Commodity coft 28 Shillings, what shall 36 Yards of the fame coft at that rate? If 24 Thus, as in Vulgar Arithmetick, if you multiply the second number 28, by the third number 36, and divide the Product by the first number 24, the Quotient will be 42 for the fourth number fought; and fo many Shillings will 36 Yards coft. CHAP. IV. Of Altimetria, Longimetria, and Planometria. Shewing how to take all manner of Heights, Depths, and Diftances, whether Acceffible or Inacceffible, and to Measure Land Mechanically. Sect. I. Of Altimetria, or Measuring of Heights. PROBL. I. How to take the Height of any Tree, Steeple, or other Upright Building, b1 the Shadow thereof. L FIG. I. ET AB be the Wall of fome Caftle or Watch-Tower, and the Sun fhining cafts the Shadow thereof upon the Level Ground to C; now having a Walking staff in your hand, fet that upright at the end of the fhadow of the Wall at C, and I find that the staff cafts its shadow to E, where I make a mark, as also another at C; then measuring the length of my staff, I find it 38 Inches, and the length of the fhadow of it CE, to be 46 Inches: Then measuring the length of the fhadow of the Wall of the Tower A C, I find that to be 30 foot, which is 360 Inches. Now for the Height of the Castle-Wall, you must work by the Rule of Proportion, thus: As CE, the length of the fhadow of the ftaff, 46 Inches, Is in Proportion to the length of the staff CD, 38 Inches; So is A C, the length of the fhadow of the Wall, 360 Inches, To297. 4 Inches, for the Height of the Caftle Wall. For For if you multiply 360, by 38, the Product will be 13685; which divided by 46, the Quotient will be 297. 4 Inches ferè, for the Height of the Caftle Wal B A, which is 2.1 Foot and 9 Inches, and fomewhat more. PROBL. II. . How to take the Height of a Watch-Tower, by the Shadow, when you cannot come to the bottom of it, to measure the length of the Shadow. FIG. II. ETAB be a Watch-Tower, whof Height I would know by the fhadow thereof, but there is a Moat about it, as B C, fo that I cannot come to meafure the fhadow thereof: However, I come near to the Moat-fide, and there I find the fhadow of the top of the Tower, to caft at C, where I erect my ftaff CG, and that cafts its fhadow to H; I meafure the length of my ftaff, and I find it 4 foot, or 48 Inches; and the length of the fhadow thereof CH, I find to be 32 Inches, thefe two I note down. 4 Then, fome time after, (when the Sun is lower) I come again to the place, and find the fhadow of the top of the Tower to caft at D, where again I erect the same staff of 4 foot long, and find that it cafts its fhadow to E, and that the length of the fhadow thereof, DE, is foot 5 inches, or 53 inches, and fomewhat better; this I alfo fet down, and then I measure the distance between the two places where the Tower cafts its shadow, at the firft and fecond time of my obfervation, namely the distance C É, and find it to be 10 foot, or 120 inches. And now having all thefe numbers fet down, I come to find the Height of the Tower A B, by help of the Rule of Proportion, as followeth. (1.) As D E, the length of the fhadow of the ftaff D F at the fecond Obfervation, 53 Inches, Is to 48 Inches, the length of the staff; So is 10 foot (or 120 Inches) the length of the fhadow between the two places of Obfervation C and D, To 108 Inches, or 9 foot. Which number 9 foot, or 108 Inches, fet down, And fay again by Proportion, (2.) As 48 Inches the length of the staff GC, Isto 10 foot (or 120 Inches) the diftance between the two places of Obfervation C and D; So is 18 Inches, (the Number before found) To 27 Inches, the Height of the Tower; which reduced into Feet is 22 Foot, 6 Inches. PROBL. III. How to take the Altitude of any upright Building, or the like, by a Bowl of Water. FIG. III. Ravelling along the Road I fee a May-pole, as KL, the height whereof I would gladly know, but having no Geometrical Inftrument, I procure a Bowl of fair Water, which I fet down upon the ground, ground at M. And then, when the Water is ftill in the Bowl, I go backward in a right line from the May-Pole, till I fee the fhadow of the top of the May Pole in the middle of the Water, which I do when I come at N; and at N, I make a mark upon the ground; then do I meafure the distance from the foot of the May-Pole at L, to the Bowl of Water at M, and find it to be 175 Inches: Alfo I measure the diftance from the Bowl of Water at M, to the place of my ftanding at N, and find that to be 72 Inches: Then I measure the Height of my eye from the Ground ON, and find that to be 60 Inches: These things known, I fay by the Rule of Proportion, If 72 Inches distance MN, give 60 Inches Altitude NO; 60 For if you multiply 175 by 6c, the Product will be 10500, which divide by 72, the quotient will be 1452, that is almoft 146 Inches, which is 12 foot 2 Inches for the height of the May-Pole K L, required. How to take the Height of any upright Building that is approachable, by two Sticks or Rulers joined together Square-wife. LE FIG. IV. ET PQ be fome Structure, ftanding upright upon plain Ground, whofe height you require. ་ Go unto fome convenient Court, Tard, Garden, or other piece of level Ground adjoining to the building to be measured, then take your Square in both your hands, holding it perpendicular, which you may do, by having a Thread and Plummet as TV, hung upon a pin near the of top the Square at T then keeping it in this pofture, go backwards or forwards (as occafion requires) till your Eye being at X, you can fee the other end of your Square at T, and the Top of the Building at P, all in one Right Line, which when you do, make a ftand, as at S; Then measure the height of your eye from the Ground XS, with a ftring, and fet that length upon the ground from the place of your ftanding at S, to R: Then measure the diftance from R to Q, for that shall be equal to the height of the building PQ, and is here 210 foot. How by the help of this Square, standing upon a Platform of a kno vn height to find the distance from the Platform to any Tree, River, or other Object that is remote from you. LE FIG. V. ET AB be a Platform, whofe Perpendicular height is 100 foot, being upon the top thereof at A, I would know how far the Oak at C, is diftant from the bottom of the Platform at B. Upon the top of the Platform at A, I erect a Pike or Javelin 12 F foot long, long, more or lefs, upon which I hang the Angle of my Square: And I look with my eye at D, along the fide of my Square, till I fee the bottom of the Oak at C, and in this pofition I fix my Square, with a Screw or the like, to the head of the Javelin: Then from D I extend a thread or Line by the fide of my Square,till it touch the Platform at E, and then I measure the distance upon the Platform from A to E, and find it to be 24 foot, 6 inches; then by Proportion I say, As 12 foot, the length of the Javelin, DA, Is to 24 foot and a half, the diftance ineafured upon the Platform A E, So is 112, the height of the Platform and Javelin together, BD, To 228 foot, 8 inches, for the distance B C. PROBL. II. How to take the distance from the place of your standing upon level Ground, to any Tree, Tower, or other thing, remote from you, tho you cannot come near the fame, by your Square. FIG. VI. Tanding at F, I fee a Conduit head at G, whofe distance from F where I ftand, I would know, but I cannot come near it for a River between Fand G: However, At F I erect a staff of 4 foot high, or 48 Inches) as FH, upon the end whereof I hang the Angle of my Square, and I look by the fide thereof, til I fee the foot of the Conduit-head at G, and fixing my Square there, I extend a line from H, by the fide of the Square, till it touch the Ground at K: Then measuring the distance between Fand K, I find it to be 3 foot, or 36 Inches: Then by the Rule of Proportion I fay, As 36, the distance K F, Is to FH, the length of the ftaff, 48 Inches; For as often as K F is contained in F H, T So often is FH contained in F G. PROBL. II. How to take the Breadth of a River by the Square. FIG. VII. Here is a River MPO, whofe breadth I defire to know: Upon the brow of the River at M, I fet up my Staff M L, which is 60 inches (or 5 foot) long, and hanging my Square upon the end thereof at L, I look by the fide thereof, till I fee the Brow of the River on the other fide at O, and there fixing my Square, l'extend a Thread by the fide thereof, from L to N, then meafuring the distance LN, I find it to be 15 inches (or 1 foot 3 inches ); then I lay by Proportion, As NM, the diftance measured, 15 inches, Is to LM, the length of the Staff, 60 inches. So is L M 6 inches, To M O, 240 inches, (or 20 fcct) for the breadth of the River MO. PROBL |