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APPENDIX;

ON THE THEORY

OF MENSURATION.

MENSURATION OF SUPERFICIES.

T

HE whole Theory of Menfuration of Superficies, confifts in finding a Rectangle equal to the given Figure.

In Theo. 17, Book ift. it is demonftrated, that every Triangle is equal to half a Parallelogram, on the fame Bafe and Altitude; and, in Theo. 18th. is fhewn and demonftrated, that, all Parallelograms or Triangles, having the fame or equal Bases and the fame Altitude, are equal. Confequently, fince Rectangles are Parallelograms (Def. 33 and 44) and it is, there, fully demonftrated that they are all equal, having equal Bafes (i. e. having any two Sides equal) and the perpendicular distance between thofe Sides and the oppofite alfo equal, it is evident, that the meafure of one is also the measure of the other. And, because Triangles on equal Bafes with Parallelograms, and being of equal height are equal half fuch Parallelograms, the Area of a Triangle is readily obtained.

Hence is deduced the general Rule (well known to all Surveyors or Artificers, concerned in measuring fuperficies) for finding the Area of any triangular plane Figure; which is, to multiply the Bafe, i. e. any one Side, by half its height, from that Side, or half its Bafe by the whole height; the reafon for which is, in thofe Theorems, clearly accounted for; and, in that confifts the whole of fuperficial Menfuration.

In Prob. 20th is fhewn how to construct a Rectangle, equal to a Triangle, on thofe Principles; and in the 22nd it is further extended to a Trapezium; or to any Quadrilatéral, whatever, egular or irregular.

The 21ft fhews how to conftruct a Parallelogram, under any Angle, equal to a Rectangle, on the fanre Bafe; and confequently, by changing the Premifes, a Rectangle may be formed equal to any Parallelogram whatever.

If the construction of thofe Figures be well understood; practical menfuration is eafily attained; which confifts (as I have obferved above) in finding a Rectangle, i. e. in knowing how to take the dimenfions of the two Sides of a Rectangle, equal to any given Figure; for, the multiplication of any two Numbers being applied to measure, always denotes, or produces the Area of, a Rectangle under fuch dimenfions; which I fhall, in the first place, endeavour to make clear and intelligible.

Most People, at the first thought on these matters, imagine, that if two Figures (of any species whatever) have equal Circuit, i. e. if the measure of all the fides, of each Figure, in one suni, be the fame, they have equal Areas; than which, nothing is more falfe, as will be made appear.

A Circle contains the greatest Area of any Plane Figure, having an equal Circumference or Perimeter. If a Circle be depreffed, though ever fo little, it becomes elliptical; in which Cafe, it lofes of its dimenfions; and the more it is depreffed, i. e. the more excentric an Ellipfis, having an equal Periphery, or Circumference with a Circle, the lefs is its Area to that of the Circle. Confequently, if a cylindrical Veffel be bulged, or it its Sides are depreffed till it becomes elliptical, it lofes of its meafure; and the more the Sides are depreffed, the more it loses; because it is evident, that, if they are preffed quite flat, it lofes the whole internal Área.

So likewife, of right lined Figures; the more the Sides are multiplied, and the nearer it approachcs, in figure, to a Circle, the greater area it contains. Confequently, regular Poligons, of a greater number of Sides, contain a greater Area, than thofe of fewer Sides and having equal Perimeters:

Hence

Hence it is plain, that a Rectangle contains a greater Area than any other Quadrilateral whatever; if the meafure of their Side, in one Sum, be equal. But, a Square, which is the most perfect Rectangle, contains a lefs Area than a regular Pentagon; and a Pentagon contains lefs than a Hexagon; a Hexagon less than a Heptagon, and fo on to a Circle, whofe Perimeters are all equal; fo that the Square and Circle, whofe Perimeter and Circumference are equal, differ greatly in Area, as fhall be illuf rated hereafter to demonstration.

Now fince, in Menfuration, à Rectangle is the standard or criterion, by which the Area of all Plane Figures, as well as all other fuperficial Contents, are afcertained; I fhall, in the first place, fhew, and account for, the methods of taking the dimenfions of various Figures; which meafures, being multiplied into each other, will give the true Area; each Figure being equal to a Rectangle under thofe Dimenfions.

1. The area of a Square is obtained, by multiplying the measure of one Side into itself.

e. g

B

2

7

E

Let the Side of the Square ABCD be five feet; the measure of a Foot being reprefented 4 by each finall divifion on its Side, as 1, 2, 3, 4, 5, 3 Now, if from thefe divifions, lines are drawn, both ways, parallel to the Sides; it will be divided into as many lefs Squares, as the Side multiplied by itself produces; viz. 5 times 5, equal 25; which, is the number of small Squares contained in the large Square, ABCD; each small square a, b, c, &c. being fuppofed one Foot, on each Side, is, therefore, called a fquare Foot. So, the Square A E, containing three feet on each Side, is a Square Yard; its Area is nine fquare Feet.

a b

с

A

2

But, when a Rectangle measures more, or is longer, on one Side than the other; then, either Side multiplied by the other, not oppofite, gives its Area or fuperficial Measure. Whereas, in a Square, the Sides are all equal; and con:equently, the measure of one is alfo the measure of any other; therefore, the Side is faid to be multiplied into itself, to produce its Area.

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2. If the Rectangle ABCD measures 8 Feet on one side (AD) and 5 Feet on the other (AB) the oppofite Sides are the fame (15.1.) and if Lines are drawn through the divifions (as in the Square) parallel to the Sides, it will be divided into 5 times 8, equal 40; the number of fquare Feet, contained in the Rectangle.

Hence, it is evident, that, in Menfuration, any two Numbers denoting Measure, in Inches, Feet, Yards, &c. being multiplied into each other, gives the area of a Rectangle under thefe Dimen fions, in fquare Inches, Feet, &c. and confequently, all multiplication of Measure, denotes the Figure, under fuch dimenfions, to be right angled. Wherefore, the multiplication of Lines, in Geometry, implies a Rectangle constructed on two Lines.

3. To any Point, E (in BC) let AE, DE be drawn, forming a Triangle, AED.

It is obvious, that the Triangle AED contains feveral whole and entire Squares, F, F, &c. and fome Pentagons, as a, a; fome Trapezia, as bb, and fome Triangles, as cc, &c.

Now, it would be no eafy matter to ascertain how many entire Squares all thofe irregular Figures are equal to; for there are but 12, entire, 4 Pentagons, 3 Trapezia, and 7 Triangles.

But, fince (by Th. 17) we have full conviction, that the Triangle AED is equal to half the Rectangle ABCD; and the Rectangle ABCD contains 49 finall Squares; confequently, all thofe irregular Figures are equal to 8 Squares; which, added to the 12 entire ones, make 20, the true Area of the Triangle AED.

4,

Next, I will fhew, that two Parallelograms, may have the Sum of all their Sides equal, and differ greatly in Area.

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But, the Area of the Par. AEFD is not equal to the Area of the Rect. ABCD. For, if AE be multiplied into AD, it will produce an Area equal to the Rect. ABCD, because, AB=AE.

But, the Par. AEFD is equal to the Rect. AGHD only.
For, it is demonftrable, that, the Triangle AGE=DHF. - 7.1.
Confequently, if DHF be taken away, and its equal, AGE, be
added, the Rectangle AGHD is equal to AEFD.

Again, if AE be produced to I, and DF to K (in BC produced)
Then, the Parallelogram AIKD is equal to the Rect, ABCD.
For, the Triangle ABI=DCK; wherefore, taking away DCK,
and adding an equal, ABI, the thing is manifeft.

Now, if the Rectangle ABCD be supposed to be depreffed, to AEFD, it is evident that it has loft of its dimenfions, confiderably; and if it be depreffed lower, to ef, it still lofes more; notwithstanding the Perimeter, A e fD, remains the fame, equal ABCD. Confequently, ABCD, contains a greater Area than any other Parallelogram, having an equal Bafe, and equal Perimeter.

Therefore, if ABCD be deprcffed, i. e. if the Angles are not Right ones, it will contain a lefs Area; for, if AB and CD deviäte ever fo little from a Perpendicular, BC muft neceffarily fall lower. Confequently, its Altitude being lefs, and the Bafe remaining the fame, its Area is lefs. Cor. 1. 6.

5. It may feem ftrange, to fome, that a Square fhould contain a
greater Area than any other Rectangle, having equal Perimeters.
Suppofe the Rectangle ABCD to measure, E

on one Side (AD) 8 feet, and on the other
B
Side (AB) 6 feet; its Arca is 8×6=48; and
its Perimeter is 8+8+6+6=28.

Let a Square, AEFH, be defcribed on AH,' equal 7 feet. Then, the measure of its Sides,' in one fum, is 7×4, or 4 times 7=28, the fame as the Rectangle.

2

A

Now, the Rectangle ABGH is common to both the Square, AEFH, and the Rectangle ABCD.

But the Rectangle, BEFG, which is the remaining part of the Square, contains 7 fmall squares; whereas, the Rectangle DCGH, the remaining part of the Rectangle, ABCD, contains but fix.

The

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