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Bearing in mind that the work is elementary in its scope, the author has left to the higher Algebra which is to form a part of this Series of Mathematical Text-books, the full discussion of many subjects that are too abstruse for the easy comprehension of beginners, and has purposely avoided the introduction of nice distinctions, which, though proper enough at a later stage in the course of study, would be likely to perplex the young student. He has preferred to use the mathematical terms positive and negative in the sense which general usage has established, rather than to attempt a limitation of their signification by applying them only to quantities to denote their essential relations, regardless of the algebraic signs which stand before them.
For the convenience of such learners as may desire to study only the most essential portions of Algebra, some topics have been excluded from the body of the work and placed in an Appendix.
The author has been led to the preparation of this Course of Mathematics by the popular and just demand for text-books which are brief and yet complete, and which are so graded and mutually consistent as to secure for the learner the greatest economy of time and labor.
He ventures to hope that the kind approval so widely bestowed upon his Series of Arithmetics may be extended to the work now presented to the public.
RTICLE 1.-1. Name some thing that can be measured. With what can it be measured?
2. What measures may be used in expressing the distance, or length, between two places? The area or surface of a farm? The capacity or volume of a room? The size of a block of stone? The time between two dates? The magnitude of a number?
3. What do you call every thing that can be measured or computed?
4. Is 6 a quantity? 5 miles? 4 bushels? Pride? Space? Redness? Length?
5. The quantity 6 apples contains how many single things? What is each single thing? The quantity 10 pounds contains how many single things? What is each single thing?
6. What is a single thing in 12 dollars? In 8 days? In 4 a's? In 3 x's? In 25?
7. What name is given to one of the things of which any quantity is composed?
8. What is the unit in 3 acres? In 5 dollars? In 11 z's? In 9 tens?
9. Express the number five by single marks; by a figure; by a Roman letter.
10. Express the number or quantity six by marks; by a figure; by Roman letters.
11. You have now expressed number or quantity in what ways?
12. In the Roman notation, what letter represents one? Five? Ten? Fifty? One hundred? One thousand?
13. Letters, then, may be used to express what? Five is frequently expressed by the letter V, but suppose we agree to express it by a, or b, or c, or any letter whatever; what then will be the value of the letter used?
14. If the value of a is five, what is the value of 2 a's? Of 3 a's?
15. If we agree to denote five apples by a and six apples by b, what will a plus b denote?
16. If we let a stand for seven, 2a for twice seven, 3a for three times seven, and so on, 10a will stand for what? 20a? 4a plus 5a? 8a plus 2a? 12a less 7a? 9a less 6a?
17. John has 4 equal piles of apples, each pile containing 15 apples. How many times 15 apples has he?
18. If we denote 15 apples by b, how many times b will denote the entire quantity of John's apples? What, then, will denote the entire quantity?
19. What will denote the entire quantity if we denote the quantity in one pile by c? By d? By e?
20. From these examples what may you infer in regard to using the same letter to represent different quantities? In regard to using different letters to represent the same quantity?
21. George tells me that he has a certain number of cents, and that his sister has twice as many, but he does not tell me the number.