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AN ANALYSIS OF TIIE SCIENCE OF NUMBERS,
WITH ESPECIAL REFERENCE TO,
MENTAL TRAINING AND DEVELOPMENT
ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIPTIVE
AND IXTEGRAL CALCULUS,
51 & 53 JOIN STREET.
BOLD BY BOOKSELLERS, GENERALLY, THROUGHOUT THE UNITED STATES
This work is designed both for primary and advanced classes The first part is adapted to beginners, while the latter part is peculiarly fitted to give to the more advanced student that cho rough mental drilling, in the Analysis of Numbers, which fur nishes the true basis of all mathematical knowledge.
It is suggested that classes in Higher Arithmetic, and eveu in Algebra, not familiar with works of this kind, will be greatly benefited by a thorough exercise in this most important branch of mathematical science.
The Teacher should require the class to dispense with their books at the time of recitation. . He should read each example, and then call upon some member of the class to solve it. The pupil should rise and repeat the example in the same language used by the teacher, and should then proceed to analyze it.
The analysis will be found to consist of three parts; two pro positions and a conclusion; thus :
What will 4 barrels of cider cost at 3 dollars a barrel ?
1st PROPOSITION: Four barrels will cost 4 times as much as 1 barrel.
2D PROPOSITION : If 1 barrel costs 3 dollars, 4 barrels will cost 4 times 3 dollars, which are 12 dollars :
CONCLUSION: Therefore, 4 barrels of cider at 3 dollars a barrel, will cost 12 dollars.
The pupil should never be allowed to omit either of the steps; and he should be required alwavs to adhere strictly to a correct and uniform phraseology in the analysis.
The forms of analysis are thought to be of great service both to the teacher and pupil.
It is also suggested, that the pupil be thoroughly drilled in Lessons III. and IV, Sect. VII, as they afford very valuable meutal exercises and a great variety of Arithmetical processes
CAJORY Entered according to Act of Congress, in the year One Thousand Eight
Hundred and Fifty-four,
BY CHARLES DAVIES. In tho Clork's Office of the District Court of the Southern Distriot of
EVERY book of instruction should have a specifio object to which the entire work, both in matter and method, should strictly conform.
It is the object of this book to train and develop the mind by means of the science of numbers. Numbers are the instruments here employed to strengthen the memory, to cultivate the faculty of abstraction and to give force and vigor to the reasoning powers.
All our ideas of numbers are either of unity or of multiplicity—unity being the elementary idea from which all others are derived.
A true analysis must conform to the nature of the subject analyzed. It must separate all the ideas and principles into their primary elements, and then explain and make manifest the laws by which these elements are connected with each other. Hence, the analysis of numbers must begin with the unit 1,-for this is the foundation, and the science is but the development of the various processes by which allo other numbers are derived from 1, as a base, and a comparison of the base 1, with the numbers so derived.
Every number has what we call a base: that is, “number being a collection of things of the same kind,” one of these things is the base of the number; and this thing, is called a unit. If we have the num
ber 3 hundred, we may consider it in several points of view :
1st. It is one hundred taken 3 times, and if we regard one hundred as the base, then, the base is taken 3 times to make up the number; and 100 is the unit.
2nd. We may consider the number as made up of 30 tens, and if we regard 10 as the base, then the base is taken 30 times; and 10 is the unit.
3rd. We may also consider the number as made up of 300 ones, in which, the base is 1, and the unit of the number 1.
Again, if we analyze the number,
cwt. gr. lb.
13 2 20 12 4 We see, that lcwt. is the base of 13cwt.; 1gr. the base of 2qr.; 1lb, the base of 2016.; lcz. the base of 12oz.; and dr. the base of 4dr., and all these bases may be referred to 1 dram as a primary base; hence, as in simple numbers, every base may be referred to the unit 1: therefore, in every entire number, 1 is the primary base.
Let us see if the same be true in fractional numbers. If we have the fractions it denotes :
1st. That something regarded as a whole has been divided into 8 equal parts: and,
2nd. That 7 of these parts are taken.
In this collection of 7 things, (each of which is }); } is the base of the fractional number ; but it is not the primary base; for implies, either of 1 or 1 of some collection of l’s; if a collection of l's we call tnat collection unity, which may be referred to the primary base 1: hence, every number, either integral or fractional, has the unit 1 for a primary base.
A fractional number, therefore, is merely a collec
tion of the equal parts of unity, and to one of these parts we give the name of fractional unit. The unit which is divided is called the unit of the fraction, and may be a collection of units, (as what is į of 40 ?) or it may
be the unit 1.
The term Unity, in mathematical science, is applied to any number or quantity regarded as a whole: the term unit, in arithmetic, to any number which is used as the base of a collection. Thus, 10 is a unit of the second order, being the base for the collection of 10's 100 is a unit of the third order, being the base for the collection of hundreds, and similarly for other bases. Thus, also, in the fraction }, is the fractional unit, being the fractional base, while the primary base is the unit 1.
Every arithmetical process, therefore, has a direct reference to the unit 1; and with this view of the subject before him, the pupil always has the means of making a correct analysis.
Addition is the process of finding a number which shall contain as many units, and no more, as are found in all the numbers added. Multiplication is taking one number, called the multiplicand, as many times as there are units in another number, called the multiplier, and the number which shows the result of such taking, is called the product: and similarly für all other arithmetical processes.
A clear conception of elementary principles, by which we mean, those principles that result from a final analysis, lies at the foundation of all knowledge, It is not till we get such conceptions, and have learned the laws by which they are connected, that we have acquired any thing deserving the name of science.