8. Six thousand five hundred and thirteen millions two hundred and eighty thousand three hundred and fourteen. millions. hundreds of thousands. O tens of thousands. thousands. co hundreds. No. 8. 6513 2 8 0 3 14 When a number is written in figures, in enunciating or expressing it in language, it is necessary, to substitute for each of the figures, the word which it represents, The following example will illustrate this: tens of billions. billions. hundreds of thousands of millions. thousands of millions. -millions. hundreds of thousands. hundreds. 2 4 8 9 7 3 2 1 5 8 0 3 4 6 It is evident that the formation of numbers by the successive union of units, is independent of the units, as appears from the above table, by means of which, we are enabled to compound and decompound numbers, which is called CALCULATION. We shall now explain the principal rules, for the calculation of numbers, without regard to the nature of their units, and proceed to Addition. Let us now, review this lecture, and enquire, what does the conjunction AND mean, when we say 4 and 5 make 9? A. Addition. Q. What then is addition? A. The collecting or putting together of numbers. Illustration.-6 and 8 are 14, which is called the sum. Q. What is the sum called? A. The amount or total aggregate. Q. What difference is there between the SUM and the AMOUNT? A. The sum means one particular sum, as 6 dollars, 8 dollars, or 10 dollars. And the amount is the sum total. These have another meaning when we speak of interest. See Interest, lecture 10th. ON ADDITION. It is a well known principle in Mental Philosophy, that the mind derives all its primary ideas from the immediate perception of the senses. A knowledge of this fact shows the importance and utility of employing sensible or tangible objects, in order to assist the juvenile capacity in comprehending the nature and combinations of abstract numbers. Nor should this be considered as a modern innovation, for we find, that the ancients frequently, had recourse to similar methods, when they attempted to teach the principles of Geometry. The Syrians, for instance, and most of the contiguous nations, looked to the Egyptians as teachers in most of the sciences. And it evidently appears, that their instructions and communications were imparted, chiefly, through the medium of hieroglyphics, diagrams, and other figurative representations, adapted to their capacities and circumstances. There is one important quality of the mind which deserves to be particularly noticed; for instance, when we speak of DOLLARS, we mean numerical increase, that is to say, ADDITION. Because, to add, is to increase from an unit. Ten thousand of these American coins may be cast in the same die, and to all appearance may be precisely alike, neither does the mind conceive them under any idea of variety, but mere increase of number; from this principle, it may be laid down philosophically, that no individual can make great advances in intellectual improvement, beyond the immediate perception of sensible objects, such as pertain, to the five corporeal senses, seeing, hearing, feeling, tasting, and smelling. This is a doctrine, to which our general train of reasoning will refer: for it is certain, there is no possibility of transmiting our ideas, but by reference, to sensible objects. The science of Arithmetic in its present structure, can be reduced to mathematical precision, its elements can be analyzed, its combinations discovered, and a basis laid for gradual improvement, according to its general and consistent laws, by Demonstrations, and Illustrations. Therefore, as the cultivation of the mind depends, chiefly on a correct habit of thinking, the teacher should, by all means convey his ideas in a plain and familiar manner, and in the commencement, have recourse to visible objects. For instance, the diagram here laid down is a plan of an orchard, there are ten rows of apple trees in it, and five trees in each row, the question is, how many trees in the orchard? Ans. 50. Again, we next introduce, the system of addition by means of the CIRCULAR DIGITS. The teacher may commence with A in the class to count, in the following 28 30 32 34 36 38 40 42 44 46 48 50 or differently, according to the numbers on the Circular Digits above mentioned. Then, let the class commence with 3, 3+3=6. 6 9 12 15 18 21 24 27, &c. up 3 3 3 3 3 3 3 3 9 12 15 18 21 24 27 30 Then again, with 4, 4 and 4 are 8, 8 12 16 20 24 28 32 36, &c. 4 4 4 4 4 4 12 16 20 24 28 32 36 40 to 51. By this exercise, in small additions, pupils can readily answer in mental operations. When the addition class is called up, each scholar should be furnished with a card, numbered agreeably to the lessons in the Addition Table. 1st card numbered the same as les.. son first; 2d card numbered the same as lesson second, and so on. Hence, it is evident, that when, the subject of addition, is commenced, the pupils become highly delighted with this instructive and amusing process; because their ideas become interchanged, between one percipient thing and another, which, no doubt, emanate, from the use of these tangible objects. 1 6 7890 The astonishing number, of changes, capable of being produced, by the various combinations of these elementary materials, afford, so many instances, of the ingenuity of man. The nine figures, with a cipher 0. give all possible pow The learner will find by adding the figures 57 laterally, diagonally, ers of numbers. 8 16 The figure A, B, C, 3 square. 4 and vertically, that the sum is 15. It appears by inspection, that the central figure 5 is a Magic number, and the favourite figure in nature, for all the machinery on earth is reducible to the following 5 distinctive forms, viz; the Wedge, Lever, Pulley, Wheel, and the Screw. |