« ΠροηγούμενηΣυνέχεια »
Fig. 104. Let the Triargle ABC be given, from whence is to be cut off in a Trapezium: Fift then on the Line CB defcribe a Semi-Circle, then divide CB into 5 equal parts, and of 3 of them from B, erect the Perpendicular DE, then fetting one Foot of your Compaffes in B, extend the other to E. which diftance fet off from B towards C, which endeth in F; by which point draw a Line Parallel to AC: So I include the Triangle GFB, to contain of the whole : And confequently the Trapezium ACFG doth contain of the fame as was required.
Firft take from the whole, the remainder will be then by the laft Problem make the Triangle GFB to contain, and then it will follow, that the Trapezium ACFG muft contain of the whole Triangle,which was required.
To divide a Triangle, when the Line of Partition goes not parallel with any fide, take this Eximple.
Fig 105. Let ABC be a Triangle to be divided into two parts which shall bear Proportion to one another, as 3 and 2, by a Line drawn from the point D in the Bife or Line AC.
From the limited point draw a Line to the Angle B; then divide the Base AC into five equal parts, and from the third point of Divifion draw the Line to E, Parallel to BD. Lastly, from E draw the Line ED, fo fhall the Trapezium ABED be in Content as three to two, to the new Triangle DEC.
I have now done with the Divifion of Triangles, when I have added thefe three Advertisements.
First, You must be fure to take very exactly the diftance of every, Point, where a dividing Line cutteth any fide, to one of the ends of the fame fide, as in the laft Figure, the distance BE and AD, which diftances being applied to the Scale by which the Figure was protracted with, will fhew at how many Chains and Links end, you are to make your dividing Line on the Field itself.
Second, The proportions by which you are to divide, are not always fo formally given as in the former Example, but are fometime to be found out by Arithmetical working, as in this Cafe. I 3
Suppose a Triangle of 6 Acres, 2 Reods, 31 Perches, must be divided, so as the one of the 2 parts shall be 4 Acres 3 Roods, and s Perches, and the other confequently 1 Acre, 3 Roods, and 26 Perches; reduce both mealnres into Perches, and the one will be 765, and the other 306. There Sum is 1671; which by their common measure being reduced into their lowest terms of Pre. portion in whole numbers, will be 5, 2, and 7, which shews that the Triangle being divided in 7 equal parts, the one must have 5 of those 7 parts, and the other 2. And obferve that it will be sufficient to find the common measure between the Sun of the Terms, and either of the Terins; the Method whereof is shewed in every Arithmatical Book for reducing Fractions into their lowest Terins.
Third, In these and all other Divisions of Land, where a striet Proportion of Quartity is to be observed, you must have refpect to the Rules hereafter delivered. But if there be any useful Pond or Well to draw the Line of Division through ; but if there be an unuseful Pond, Lake or Puddle; or it there be any Boggy or barren Ground, that must be cast out in the Division
; meafure that first, and fubftract it from the Content of the whole Close, and then lay the juft Quantity of the remainder on that fide that is free from it, that the other inay have
his juft part alfo, befides that 'which is ufelefs.
To cut off from a Square any part propounded in a Parallelogram.
Divide the parts to be cut off, by the fide of the Square; the Quotient fhew's how much of the fide of the Square you fhall take for the Breadth of the Parallelogram; at which diftance draw a Parallel Line, which shall include the Parallelogram required.
Fig. 106. Admit ABCD to be a Square iven, whofe fide is 20; from whence 160 parts is to be cut off with a Line Parallel to CD, fo then CD makes one of the fides of the Parallelogram; then work as is before taught, and draw the Line EF; which includeth the Parallelogram (DEF, and contains 10, the parts required.
Note, And if you would have cut off, ,&c. then you must divide the Square Side to be cut off into thefe Proportional parts, and fo by thofe parts draw a Parallel Line, which would have included a Parallelo
gram to have contained the parts Propor tionable.
From an Irregular Figure to cut off any parts required.
Meafure fo many Triargles lying next to the part affigned, till you have fomething too much, then by the firft and fecond Problem of this part, cut of the overplufs from the laft Triangle, fo fhall you have a Figure to contain the parts required..
Fig. 107. Admit ABCDEFG to be a Plot, from whence 480 parts are to be cut off with a Line iffuing from C, and to be tcwards the fide AB: Firft, let the Trapezium ABCG be menfured, whofe Area is 370, which added to 136, the Area of the Triangle CFG maketh 506, which is too much by 26, which 26 let be cut off from the Triangle CFG, by the Line Cb; fo doth the Figure ABC b G, contain 480, the parts res quired to be cut off.
Note, This Problem is very material in the Practice of Surveying, in dividing and