SQUARE MEASURE

237. The square vara (sq. vr.) is the unit.

NOTE.-The term league usually means 4428.4 acres of land without regard to shape, or in Spanish, a sitio de ganado mayor (a site for large cattle). There is also a sitio de ganado menor (a site for small cattle), which contains 11111111+ sq. vr., or 1968.1777 acres. The expression "large cattle" has reference to horses, cows, etc., and "small cattle" has reference to sheep, goats, etc. "Cattle" is used in the sense of "live stock."

The term labor is now applied to 1000000 sq. vr., or 177.136 acres of land, but the original idea was that the labor was a field for farming purposes attached to the league, which was used for pasture land.

The porcion (share) has no fixed size. Along the Rio Grande in Texas, a porcion usually contains 30000000 sq. vr., or 5314.08

acres.

Exercise XXXI

1. What is a vara? A labor? A league?

2. How many varas in 100 yards? In 320 rods?

3. How many varas in 8 miles?

4. Reduce 11404.8 varas to miles.

5. The distance from Austin to San Antonio is 152064 varas; find the distance in miles.

6. The distance from Fort Worth to Dallas is 30 miles; find the distance in linear leagues.

7. How many square yards in a labor?

8. How many varas in one side of a square league?

9. How many miles of fence does it require to inclose a square league of land?

10. How many acres in a rectangular field 684 varas in length and 345 varas in width?

11. Find the number of square varas in a ranch containing 30000 acres.

13. Find the number of sq. varas in a park 4876 feet in length and 3764 feet in width.

13. Find the number of square feet in a lot 98 varas in length and 56 varas in width.

14. Find the number of square leagues of land in a county 30 miles square.

15. Verify the following rules:

(1) To reduce varas to feet, multiply by 100, then divide by 36.

(2) To reduce feet to varas, multiply by 36,

(3) To reduce sq. vr. to acres-multiply by 177.136, then divide by 1000000.

(4) To reduce acres to sq. vr.,-multiply by 1000000, then divide by 177.136.

CHAPTER X

MENSURATION

238. Mensuration (Latin mensura, a measure) is that branch of mathematics which treats of geometrical magnitudes. It is the application of arithmetic to geometry.

239. The four fundamental geometric concepts are the point, the line, the surface, and the solid. All but the point have magnitude.

1. A point is that which has position, but no magnitude.

2. A line is that which has only length. If a point moves, it generates a line.

3. A surface is that which has only length and width. If a line moves (not along itself), it generates a surface.

4. A solid is that which has length, width, and thickness. If a surface moves (not along itself), it generates a solid.

I. PLANE FIGURES

240. A plane is a surface such that any two points in it can be joined by a straight line which lies wholly in the surface.

241. A plane figure is any portion of a plane bounded by lines.

242. A polygon is a plane figure bounded by straight

lines. The perimeter of a polygon is the sum of the lines bounding it.

243. The least number of straight lines which can inclose a plane is three.

A polygon having three sides is a triangle; four sides, a quadrilateral; five sides, a pentagon; six sides a hexagon, etc.

244. A regular polygon is one whose sides and whose angles are equal.

245. The diagonal of a polygon is the straight line joining two angles not adjacent.

246. The base is the side upon which a figure is supposed to stand.

247. The altitude of a polygon is the perpendicular distance from the highest point to the line of the base.

248. The center of a regular polygon is the point within the polygon, equally distant from the middle points of the sides; and the apothem of such a polygon is the perpendicular line drawn from the center to the middle of a side.

249. A parallelogram is a quadrilateral whose opposite sides are parallel.

250. A rectangle is a parallelogram whose angles are right angles.

251. A square is a rectangle whose four sides are equal.

252. A trapezoid is a quadrilateral with only two sides parallel.

253. A trapezium is a quadrilateral with no two sides parallel.

254. A circle is a plane figure bounded by a curved line, every point of which is equally distant from the center. The boundary line of a circle is called the circumference. The diameter is a straight line drawn through the center of the circle, its

end points being in the circumference. The radius is a straight line drawn from the center to the circumference.

255. The area of a plane figure is its amount of surface. Area is denoted by the number of square units a figure contains.

256. The unit of surface is a square whose edge is a linear unit.

257. All surface measurements are based on the rectangle.

258. The fundamental principle is:

The area of a rectangle is (abstractly) equal to the product of the base by the altitude.

ILLUSTRATION

Find the area of a rectangle 5 in. long

and 3 in. wide.

3 in.

5 in.