REMARKS. HAVING for many years witnessed the necessity of a practical elementary treatise on Geometry and Mensuration in our DistrictSchools and Academies, and as no work answering this description is at present before the public, this, it is presumed, will be a sufficient apology for the attempt to supply a vacancy in this department of science. Next to a correct knowledge of Arithmetic, we may rank that of Mensuration, a branch of mathematics which embraces a variety of pleasing and interesting subjects of the greatest importance and utility to the man of business, the merchant, arti san, manufacturer, farmer, mechanist, and indeed almost every i person whose transactions are of sufficient importance to require the application of figures. Even its introduction into “ Female Seminaries” should not be considered as derogatory, or “ too rude” for the exercise of the female mind, as a substitute for less useful studies and employment, when it is known that many ladies have distinguished themselves, as much by their attainments in the abstruse sciences of mathematics and astronomy, as they have by their contributions to our periodical literature. The daily and constant intercourse of men, in the transaction of business, will continually demand the application of the principles of Mensuration, almost as frequently as that of common Arithmetic, and notwithstanding the demand for the application of those rules and principles, it is believed that not more than one person in fifty is even partially acquainted with the science. This practical part of mathematics has been quite too much neglected in our district schools as a part of elementary instruction, and seldom receives that attention necessary to acquire a correct knowledge of the science even in many of our “High Schools and Academies.” This may be attributed either to the want of suitable books, or to the neglect of the teacher. In a work of this kind, but little new or original can be written or expected, beyond the arrangement, the selection of appropriate practical questions, and the general adaptation of the volume to the use of the man of business, and the capacity of those for whom it is intended as an introductory course of instruction to the higher branches of mathematics. In the selection of suitable matter in the following pages, much care and attention have been bestowed, with a view to introduce such, and such only as appeared the most useful, and to omit all extraneous subjects as irrelevant in a text-book on elementary science, as well as a fruitful source of complaint with many teachers. A series of examples in Decimals and Evolution has been introduced as a preliminary course, a correct knowledge of which is indispensable in the solution of the following problems, for the reason, that this part of Arithmetic is but partially, or imperfectly taught by authors or teachers, particularly decimals, they being considered of " no kind of use," and the time of the pupil is therefore employed on "mental arithmetic," and "foreign currency,” much to the detriment of the pupil, injustice to the public, and a hinderance to the advancement of mathematical science and general information: a system that never has been, nor never can be attended with the least benefit to any one, and should be prohibited and discountenanced in every school in the Union. This no doubt will be the case when the “public mind” becomes enlightened on the pernicious consequences, and convinced of the folly of a superficial education, and the great loss of time and money in the vain pursuit of a phantom,—« in truth, a most consummate humbug." This volume, with the two “ Columbian Calculators,” embraces about FOUR THOUSAND TWO HUNDRED questions for solution, a large proportion of which are original, a number sufficient to make the pupil a proficient in the various subjects introduced. If this volume should merit and receive the approbation of TEACHERS and others interested, the author will consider himself sufficiently rewarded for all his care and labor, and his most sanguine expectations fully realized. SIGNS AND TABLES USED IN THIS WORK. 1. ARITHMETICAL SIGNS. + Plus, or more. Sign of addition, as 8 + 4 = 12; 8 and 4 are 12. · Equality, or equal to. As 3 feet added to 3 feet = 6 feet. - Minus, or less. Sign of subtraction; as 8. 4= 4; 4 from 8 and 4 remain. X Into, or by. Sign of multiplication, as 4 X 2 = 8; 4 multiplied by 2 is 8. • Divided by. Sign of division; as 25 ; 5 = 5; 5 is contained in 25 42 25, 5 times, or, - 7. 6 16% 3 Cube root, 3/27 3, or 27: 3. » Biquadrate root, ^ 16 = 2, or 16+ = 2. = 4. 66 2. LAND, OR SQUARE MEASURE. 144 inches make 1 sq. foot. 9 feet 1 sq. yard. 36 feet 1 sq. fathom. 2727 feet, or 304 yards, make 1 sq. rod, pole,' or perch. 1600 poles 1 sq. furlong. 40 poles 1 sq. rood. 4 roods 1 sq. acre. 160 poles, or 4840 yards “ 1 sq. acre. 640 acres (1 section) 1 mile. 3. DISTANCES, OR THE CHAIN. 78% inches make 1 link. 25 links 1 pole. 100 links 1 chain. 10 chains (of 4 poles*), make 1 furlong. 8 furlongs 1 mile. 5 rods, or 66 ft., or 100 links, make 1 Gunter's chain. 80 chains, or 320 po. (in length) 1 mile. 10 square chains 1 acre. * Some surveyors use the chains of 2 poles, others of 4 poles. 7 4. SOLID, OR CUBICAL MEASURE. By this are measured all things that have length, breadth, and thickness. 1728 inches make 1 foot. 27 feet 1 yard. 40 feet of round timber, or 50 feet of hewn timber make 1 ton. 42 feet make 1 ton of shipping. 16 cubic feet make 1 foot of wood, or one cord foot. 128 solid feet, or 8 feet in length, 4 feet in breadth, and 4 ft. in height, or 8 X 4 X 4= 128 = 1 cord of wood. 66 5. LONG MEASURE. 12 inches make 1 foot. 3 feet 1 yard. 55 yards, or 161 feet make 1 rod, pole, or perch. 40 rods 1 furlong. 8 furlongs 1 mile. 3 miles 1 league. 60 geographical, or 693 statute miles 1 degree. 360 degrees a great circle of the earth. 4 inches make 1 hand. 9 inches = 18 inches make 1 cubit. 6 feet 1 fathom. 1 span. 66 6. OTHER MEASURES. 282 cubic inches 1 gallon, ale measure. 231 1 gallon, wine measure. 2684 1 gallon, dry measure. 244 cubic feet, or 16 feet in length, 11 in breadth, and 1 foot in height 1 perch of stone. (16.5 = 2 = 8.25 + 16.5 = 24:75.) 2150-4252 cubic inches = 1 bush. strick measure. 2553.6299 1 bush. heaped The Winchester bushel is 18.5 inches in diameter, and 8 inches deep. 66 For convenience, the following table may be used. SOLID, OR CUBIC MEASURE. Cubic Inches. Cubic Feet. Cubic Yard. Cubic Cubic Cubic Furlong. Mile. 1728= 1 46656 27 1 7762392 44823 166.375= 1 496793088000 287496000 10648000 64000 2544358061056000/147197952000 5451776000 32768000 INTRODUCTORY COURSE TO MENSURATION. REMARKS. It is presumed that the pupil is already well acquainted with common arithmetic; for, without a correct knowledge of that important science, he cannot reasonably expect to make much proficiency in the higher branches of mathematics, satisfactory to himself or creditable to his instructor. But, as this may not have been attended to with sufficient care, there can be no objection to a cursory review of the most important rules used in the solution of questions in mensuration. As it is foreign to the subject to enter into an arithmetical demonstration, the several rules will first be given, as a reference, and the questions for solution follow promiscuously, as this method will have a tendency to familiarize the pupil in their use and application. DECIMAL FRACTIONS. ADDITION. RULE.—Write the numbers under each other, observing to place tenths under tenths, hundredths under hundredths, &c. Be particular that the decimal points stand directly under each other, in a perpendicular line, both in the given numbers and in the sum or amount. Then perform the operation the same as in addition of integers. SUBTRACTION. RULE.—1. Write the numbers the same as integers, observing that the decimal points stand directly under each other. 2. Then subtract the same as in whole numbers, and place the decimal point in the remainder under those above. |