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Find the value of 1767 tons 17 cwt. 1 qr. 19 lbs. 7 oz. 14 – drams at £648 14s. 92d. + of farthing.
£1,146,891 104 If I have an eighth of a fifth part of 20001., what will
1 x 1 be the value of my share ? $ of }
= to 20001.
8 X 5 - 40 = 501.
Required the value of two ninths of 15l. ? 151. x2= 30 • 9. Required sum is 31. 98. 8d.
A package of goods weighs one ton; what will be the weight of two elevenths of it? 1 ton 20 cwt. X 2 = 11 x 4 x 28 x 16 x 16. Ans. 3 cwt. 2 qrs. 15 lbs. 4oz. 54 dr. is the value required.
A person possessed of işths of a coal mine, sells ths of his share for 20001., what is the whole mine worth? of _ == Vis is ths of the mine is worth 20001.,
20001. by the question .. sth is worth : 6661.138.4d.
3 ... the whole mine is worth ten times this amount or (6661. 138. 4d.) x 10 = 66661. 138. 4d.
What do you mean by the square of a number? If a number be multiplied by itself, the result is its square.
Does squaring a number add to or diminish its value ? It depends on circumstances: a whole number is increased by squaring it, but a fraction is diminished. The square of 4 is 16, and of 4 is 7o.
Give a method of squaring mentally all numbers below 100. Add to the number as much as will make it terminate in a cipher; subtract from the number as much as you have added, multiply the two results together, and add the square of the number added, the sum will be the answer: Thus, what is the square of 26 ? To 26 add 4, we obtain 30; from 26 subtract 4, and we get 22; now 22 x 30 = 660, and 42 = 16, hence the square of 26 = 676.
This operation seemis tedious, but a little experience will cause it to be done with very great facility. When the number to be squared ends in 5, as 25, the process is much simpler. Add 1 to the first figure, and it becomes 3, and 3 x 2 = 6 and 5 X 5 = 25, the result is 625; 45 is similarly squared 4 +1=5,5 x 4= 20, the two first figures of the result, and 5 X 5 = 25; hence the square of 45 is 2025.
ALGEBRA. 1. Arithmetic is that branch of Mathematics which treats of the properties of numbers, and the method employed for performing calculations by means of characters or figures representing particular numbers. Universal Arithmetic or Algebra is an extension of the principles of arithmetic, in which general characters, particularly the letters of the alphabet, are employed to represent all quantities which can be numerically compared with one another, whether these quantities be given or required.
2. The term Quantity is employed to denote all real objects, such as lines, surfaces, solids, as well as all other objects or qualities which may be shown to possess numerical relations. Thus weight is a quantity, because one body may be twice or three times as heavy as another. Heat is a quantity, because we can say that the temperature of the air in winter, for example, is 32 degrees, and in summer 60 degrees or upwards.
3. In order to compare quantities with each other it is necessary to fix on some determinate quantity of the same kind which we consider as our unit of measure. Thus if we wish to measure the distance between two places, we adopt an unit of length, as an inch, a foot, a yard, a mile, the distance being expressed by so many inches, feet, miles.
If we wish to compare the eights of two or more bodies we fix on a unit of weight, as a grain, an ounce, a pound, the weight of each body being expressed by so many grains, ounces, pounds.
4. In all arithmetical operations then, as addition, subtraction, &c., the pupil must bear in mind that the quantities which enter into his investigations may always be expressed by numbers, the same unit being used for all quantities of the same kind.
5. The term mathematics is employed to denote that immense and invaluable collection of human knowledge which treats of all the properties and relations of objects, whether real or only conceived to exist, which can be numerically compared with each other. The two great divisions of mathematical science are Arithmetic and Geometry. Under the first may
comprehended algebra, and the differential and integral calculus or fluxions ; under the second, plane trigonometry, conic sections, and spherics.
6. The pupil will not therefore be surprised when he is told that he is not now beginning the study of mathematics, but that in fact he commenced that study the moment he began to learn the addition or multiplication table. We
e may also inform him that if he has . acquired the common principles of arithmetic with any. thing like a clear understanding of the subject, he will find comparatively little difficulty in mastering all that is contained in this volume.
7. The object of every process of reasoning, or calculation, or investigation is to discover, from knowing the values of certain quantities and their relations to others, quantities whose values were previously unknown. But though this be our ultimate object, the inquiry often terminates when we have discovered general properties which will enable us to ascertain any numerical results in particular cases when these results
be required. Thus, for example, in common addition we have given several numbers to determine one which shall be equal to the whole taken together. Or we may be required to investigate a rule for finding a fourth, proportional to any three given numbers.
8. Mathematics are also divided into pure and mixed or applied mathematics. Pure mathematics treat of the relations of quantities conceived to exist, separated or abstracted from all natural objects. Mixed mathematics apply the abstract properties previously investigated, to the solution of problems or otherinvestigations connected with the real objects which we find in nature. Pure mathematics are necessarily and eternally true. Mixed mathematics are true only with regard to the nature and properties of bodies, such as we find them to exist, which, for anything we know to the contrary, might have been different.
When numbers are applied to denote any particular objects, they are commonly called concrete numbers, as 5 feet, 3 shillings, 4 ounces, fc. When we merely consider the characters as equally applicable to all objects they are called abstract numbers. Thus, 2, 3, 5, &c. are abstract numbers. With examples in pure mathematics the learner is already familiar by his knowledge of the abstract properties and relations of numbers, lines, surfaces, fc. We shall give him an example of mixed or applied mathematics. Example:
A philosopher meeting with a false balance (or one having one of its arms longer than the other), and wishing to ascertain the weight of a piece of metal, weighed it in one of the pans and found it to be 9 ounces, and then in the other, and observed that it weighed only 4 ounces : required its real weight. When the pupil is further advanced, he will be able to solve this question, and find the real weight to be 6 ounces.
9. When the object is to arrive at a particular result for a particular question, or a general result for a general question, the thing proposed is called a problem ; and the working out of the whole, the solution of the problem.
When the object is to show that the statement or enunciation of a particular property is true, the property so stated is called a theorem; and the showing or proving the truth of the theorem, is called its demonstration.
Example of a problem: Divide 20 into two parts which shall be to each other as 2 to 3.
Example of a theorem : Prove that any number divided by 9 will leave the same remainder as the sum of its digits or figures divided by 9.
10. The investigation or solution of a problem may either be conducted by synthesis or analysis. When conducted by the first method, we proceed step by step from first principles till we arrive at the result which was the object of our inquiry. When the process is conducted by analysis, we make no distinction between