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87. A gig and two horses cost together £75'; the first horse and the gig together cost £55; the second horse and the gig together cost £50'; what was the cost of each?

88. A man engages to do a piece of work in 20 days for £5; but after doing of it, he finds that, in order to complete the work in the time, he must call in the assistance of another workman of skill and activity equal to his own. How long must this assistant work, and what amount will be due to each?

89 A quantity of corn, which would suffice for a horse, a mule, and an ox 8 days, would be consumed by the horse and mule together in 12 days, but would last the mule and ox 16 days. How long would such a quantity serve each animal separately? 90 A, B, and C, sharing the profits of an adventure, find that the Sum of A's and B's gains is £13, that of B's and C's £123, whilst A's and C's shares, together, amount to £1133. What was the gain of each?

SUMMARY OF PRINCIPLES.

PRINCIPLE I. The same Digit is made to represent different Numbers by changing its place, counting from the Decimal Point. Page 3.

PRINCIPLE II. The Cipher is employed to fill places where Significant Figures are not required; and the Cipher is only of service when it stands between a Significant Figure and the Decimal Point. Page 3.

PRINCIPLE III. There are as many Units in the Half of a Number as there are twos in the Whole of it; as many Units in the Third as threes in the Whole, &c. &c. Pages 23 and 24·

There are as many times

in one Half of any Integer) as there are Units in one Third

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in one Fourth

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in the whole of that Integer. (P. 101·)

COROLLARY. Any one part of an Integer is at once expressed by writing the Denominator of that part under the Integer. Page 100.

PRINCIPLE IV. In the Decimal System of Arithmetical Notation, each Digit has one tenth of the value it would have in the next place to the left, and tenfold the value it would have in the next place to the right. Page 26.

PRINCIPLE V. Unlike Units cannot be compared in any way. Page 35.

COROLLARY. There is no Ratio between unlike Concrete Quantities. Page 146.

PRINCIPLE VI. The Multiplicand and the Multiplier may be exchanged the one for the other, and no alteration will be made thereby in the Product.

PRINCIPLE VII. In whatever order Numbers are multiplied together, their Continued Product is always the same. Page 53.

PRINCIPLE VIII. Multiplying by any power of 10. merely removes every Digit of the Multiplicand as many places to the left as there are Ciphers in the Multiplier. Page 55.

PRINCIPLE IX. The word "of" between Numbers has the same meaning as the sign of Multiplication. Page 56.

PRINCIPLE X. Multiplying by Digit 1, standing in any place, merely removes every Digit of the Multiplicand to the right or left, as many places as the Multiplying Digit is from the Units' place. Page 60.

PRINCIPLE XI. The Product of any Number and a Composite Multiplier is equal to the Continued Product of that Number and any set of Corresponding Factors of the Composite Multiplier. Page 64

PRINCIPLE XII. Multiplying by a Single Digit, standing in any place, removes each Figure of the Product, to the right or left, as many places as the Multiplying Digit is from the Units' place. Page 66.

PRINCIPLE XIII. Multiplying or Dividing the Numerator and Denominator of a Fraction by the same Number makes no alteration in the value of that Fraction. Pages 79. and 134.

PRINCIPLE XIV. Each Unit of the Quotient is a Complex Unit, and is equivalent to the whole Divisor. Page 99.

PRINCIPLE XV. The Quotient is at once expressed as a Vulgar Fraction, by placing the Divisor under the Dividend. Page 99.

PRINCIPLE XVI. Any Fraction is the same part of its Numerator that Unity is of its Denominator. Page 101. PRINCIPLE XVII. Dividing by any Number produces the same figures as Multiplying by the Reciprocal of the Divisor. Page 106.

PRINCIPLE XVIII. The Dividend and the Quotient in

crease and decrease together. Multiplying the Dividend (before Dividing,) also multiplies the Quotient: and previously Dividing the Dividend also divides the Quotient of the subsequent Division. Page 107.

Or, Division followed by Multiplication produces the same result as Multiplication followed by Division; supposing, of course, the Divisor and Multiplier to remain unchanged. Page 151.

PRINCIPLE XIX. The Quotient and Divisor are Corresponding Factors of the Dividend. Page 112.

COROLLARY I. A Fraction multiplied by its Denominator gives its Numerator. Page 112.

COROLLARY II. The Divisor and Quotient may exchange places, if Dividend be retained unaltered. Page 113. PRINCIPLE XX. Any Number divided by one of its Factors gives its Corresponding Factor. Page 113.

PRINCIPLE XXI. Dividing by a Single Digit, standing in any place of Integers or Decimals, removes every Quotient-figure as many places, BUT IN THE CONTRARY DIRECTION, as Dividing Digit is from Units' place. Page 128:

EXTENSION OF PRINCIPLE XXI. Dividing by any Integer or Decimal removes every Quotient-figure as many places, BUT IN THE CONTRARY DIRECTION, as righthand Significant-figure of Divisor is from Units' place. Page 129.

PRINCIPLE XXII. Every Measure of any two Numbers measures also a. Their Sum. b. Their Difference. c. Any Multiple of either of them. Page 132.

PRINCIPLE XXIII. When four Numbers are Proportionals, the Product of the Extremes is equal to the Product of the Means. Page 148.

PRINCIPLE XXIV. If the same operation be performed upon equal quantities, the results will be equal. Page 160. PRINCIPLE XXV. Any Multiple of a Quantity is equal to the Sum of such Multiples of all the parts of that Quantity. Page 157.

PRINCIPLE XXVI. The Quotient of the Sum of any set of Numbers by any Divisor is the same as the Sum of the Quotients of the separate Numbers by the same Divisor. Page 157. Rule III., for removal of Vinculum.

END OF PART I.

THE CONSTRUCTIVE ARITHMETIC.

PART II.

APPLICATION OF PRINCIPLES AND ELEMENTARY PROCESSES TO COMMERCIAL ARITHMETIC.

MEASUREMENT OF CONCRETE QUANTITIES.

THE various kinds of Concrete Quantity with which Arithmetic has to do, are: I. Time; II. Lineal Distance, consisting of Length only; III. Superficial Extent, or Area, made up of Length and Breadth only; IV. Solidity, having Length, Breadth, and Thickness; V. Capacity, or power of containing, depending on Length, Breadth, and Depth; VI. Weight; VII. Money-Value.

Every Concrete Quantity must, of necessity, be measured by a Unit of its own nature. Principle v.

We ascertain the magnitude of any given Length by trying how often it contains some settled Unit of Length. Duration can only be measured by a Unit of Time; Surface or Area, by a Unit of Surface, or a Superficial Unit. Solidity and Capacity are measured by Solid Units; Weight by Units of Weight; and Value by Units of Value.

The measuring units are called Units of Comparison.

Many of our Concrete Units of Comparison, having been selected in half-barbarous times and by illiterate persons, originated from very rude and clumsy contrivances. For example, our smallest Unit of Length was, until very recently, called a barley-corn, and is said to have originally been, really, the length of "a grain of ripe barley taken from the middle of the ear." The inch was the length of three such grains laid end to end.

The yard was the length of king Henry the First's arm; and, to the present day, a sailor roughly measures a fathom of rope by stretching it across his breast to the full extent of his arms.

The foot, the hand, the span, and the pace, were, as their names imply, similarly derived. So was also the weight called a grain. The ancient Unit of Length, called a cubit, was originally the length of a man's forearm and hand, that is to say, from the elbow

to the tip of the middle finger; and this was held to be the fourth part of a well-proportioned man's stature.

IMPERIAL STANDARD MEASURE.

It was enacted in 1824, and again in 1855', that all our legal Units of Length shall be tested by the Imperial Standard Yard; which is the distance, in a straight line, between the centres of the two gold plugs or pins in the bronze bar deposited in the office of the Exchequer. The foot is one third of this yard, and the inch is one thirtysixth of it, or one twelfth of the foot.

METHOD OF RESTORING THE STANDARD.

As it is quite possible that the Imperial Standard Yard, and all certified copies of it might be lost or defaced, it was necessary to provide a method by which the Standard of Length might be restored by reference to some natural unvarying unit.

The Act of 1824 provided that the natural Unit of Length to which we should refer, in order to restore our present system of Measures, should be a Pendulum † vibrating seconds of mean time in vacuo, at the level of the sea in the latitude of London.

By dividing such a Pendulum into 391393 equal parts, and then taking 10000 of those parts, we should obtain the length of our present inch, which is, consequently, % of the length of the Pendulum in question. And the Pendulum is

391397

=

39.1393 inches.

But, as the said Pendulum must be of the length to vibrate seconds, it becomes first necessary to establish the duration of a second. Hence, the Unit of Time lies at the foundation of our whole system of Concrete Units of Comparison. And these are successively derived in the following order, which therefore indicates the natural arrangement of the Tables of Measures and Weights.

I. Time.

II. Lineal Measure.

III. Superficial Measure.

IV. Solid or Cubic Measure.

*See note to Page 180

V. Capacity, or power of con-
taining.
VI. Weight.
VI1. Money-Value.

+ A Pendulum is a rod swinging freely on a fixed point, and it has been proved that every Pendulum vibrating in any certain time, (a second, for example,) will be of the same length, provided

I. That it meet with the same amount of resistance from the air.

II. That it be situated at the same height above the level of the sea.

III. That it remain in the same latitude.

The last two conditions amount to this one, namely, That it be at the same distance from the earth's centre.

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