the Roman abbreviation IS.,— usually written HS.,― of the Latin word sestertices, meaning two asses and a half, the two I's standing for the number two, and the S for the word semis that is semi as, or halve an as, a brass coin nearly equal in value to a dollar and a half of our money. The sestertices was a silver coin worth a little more than three dollars and a half. "The as was not a brass, but a copper coin. Smith, Dictionary of Antiquities, says, "The Romans had no other coinage except copper or bronze, till B. C. 269, when silver was first coined." The value of the as varied at different periods, but I am not aware that it was ever more than two cents of our money. In the time of Cicero, it was worth about a cent and a half. The sestertices was not worth three dollars and a half, but about three cents and nine mills.

W. C. C.

IS IT PROPER TO DIVIDE CONSEQUENT BY ANTECEDENT TO OBTAIN THE RATIO OF THE LATTER TO THE FORMER?

BY W. D. HENKLE, LEBANON, OHIO.

THIS question has been answered in the affirmative by Davies, in his Logic of Mathematics and in his Mathematical Dictionary. He says that he has adopted this method "only after the most careful consideration of the arguments existing in favor of each."

His first argument is, that "This use of the term ratio is perfectly consonant with its employment in ordinary language," as when it is said that the population of the country is increasing in a rapid ratio."

rect.

His idea of what is meant by this expression is manifestly incorThe United States' censuses do not give populations that form a progression in which the ratio is continually increasing, as he explains, " in a rapid ratio." The ratio of the annual increase to the population at the beginning of each year is slightly variable, but it is neither continually increasing nor continually decreasing. The expression means that the annual excess of births over deaths is rapidly increasing. The word ratio is not used at all in its

mathematical sense, unless we admit the old use of the term for common difference. Indeed, I think that in ordinary language, we hear "at a rapid rate" rather than "in a rapid ratio." Even if we admit Davies's explanation, we lose nothing, for by the ratio of the populations at any two epochs we mean the ratio of the population at the latter epoch to that at the former. So, too, when we say "that the units of our common system of numbers increase in a ten fold ratio," we mean that the ratio of any denomination to the one next inferior is always ten.

Again, that "In comparing numbers, the mind necessarily fixes upon 1 as a standard," proves nothing, for in conceiving of the relation of 6 to 1, we fix upon 1 as the standard, and 1 is the consequent of the ratio 6:1.

His next argument is drawn from the Rule of Three. "In order to find the fourth term, we have only to multiply the third by the ratio of the first to the second." "This simple rule for finding the fourth term, cannot be given, unless we define ratio to be the quotient of the second term divided by the first."

Was Dr. Davies so short-sighted that he did not see that the word divide might occupy the place of "multiply" in his " simple rule"? Cannot the fourth term of 6:2::9:x be found just as easily by dividing 9 by 3, the true ratio of 6 to 2, as by multiplying 9 by, or Davies's ratio of 6 to 2?

"A concrete quantity can only be expressed numerically by the quotient obtained by dividing such quantity by its unit of measure, whatever that unit may be.

In the expression 6 dollars, 6 in abbreviated language is the ratio of 6 dollars to 1 dollar, and not of 1 dollar to 6 dollars, as Davies would have it. To be consistent, he would, in finding the relation or ratio of a crib of corn to a bushel measure, have to divide the measure by the crib.

"The adoption of this definition insures uniformity," for "all writers concur in regarding the ratio of a geometrical progression as the quotient of the second term by the first."

This use of the word ratio is legitimate, for by it is meant the ratio of any term of the series to the preceding term. The defininition of the ratio of a geometrical progression as the ratio of any term to the preceding one, is given by Briot, Bourdon, Hind, Do

charty, and Dodd. Peacock, Darley, and Byrne agree in saying that the ratio of a geometrical series is the inverse ratio of any two consecutive terms. Hence we see that Davies is Hence we see that Davies is wrong in charging authors that divide antecedent by consequent to express ratio with departing from this method in speaking of the ratio of a geometric series, and "without any explanation of a change in the definition." They do not depart from their definition, and the seven authors above referred to have brought out expressly the fact that the word ratio is used in no new sense.

Davies closes his arguments with, "The considerations of analogy, convenience and uniformity, taken together, leave no room for the adoption of a contrary definition." I accept this assertion for his definition of ratio is "contrary" and has nothing to recommend it except its contrariness. The attempt to give currency to an interpretation of ratió directly the opposite of the prevailing and timehonored one, on such flimsy arguments, is unjustifiable.

Some may be disposed to consider it a very small matter whichever way may be adopted in the abbreviation of a ratio. This is a mistake, as will appear in the following reasons for adhering to the old method.

First. It is philosophical. The only proper answer to the question, "What is the ratio of 6 to 2?" is, 6 is 3 times 2, and not 2 is of 6 as Davies would have it. The direct answer to the question "What relation did Sir Wm. Herschel bear to Sir John Herschel?" is, "Sir Wm. Herschel was the father of Sir John Herschel," and not "Sir John Herschel was the son of Sir Wm. Herschel." (Let it not be understood that I intend to convey by the form of this question that both of the persons are dead.) These illustrations show the order of the thought in the idea of ratio or relation, and that the new method is an inversion of it. This fact alone settles the question.

Second. This method is used in nearly all works upon pure and applied mathematics, and, therefore, should not be changed except for strong reasons.

Third. A change would necessitate an alteration in many prevailing expressions. The ratio of the circumference to the diameter would no longer be 3.1416; the differential coëfficient would no longer be the ratio of the increment of the function to that of

the variable; the sine would no longer be the ratio of the perpendicular to the hypothenuse; the specific gravity of a substance would no longer be the ratio of its weight to that of an equal bulk of water; the velocity would no longer be the ratio of the space to the time; the probability of an event would no longer be the ratio of the favorable cases to the whole number of possible cases; the ratio of the sun's influence upon the tides to that of the moon would no longer be one-half (.448); the ratio of the number of vibrations in the octave to the number in the key-note would no longer be 2; etc., etc.

Fourth. This method is in harmony with: as a sign of division, this sign being an abbreviation of ÷ or at least equivalent to it.

So natural is the correct method of expressing ratio that frequently good is present with those who would do evil. Even the great apostles of the false method have occasionally acted righteously. Lacroix in his " Géométrie" calls 3.1415926 the ratio of the circumference to the diameter. Davies says, Pract. Math., p. 234, 1846, "The specific gravity of a body is the relation which the weight of a given magnitude of that body bears to the weight of an equal magnitude of a body of another kind." Davies's ratio applied to this definition would make the specific gravity of gold one. nineteenth.

NOTE. Most writers, including those who adopt the false ratio, define a geometric series as one in which the terms increase or decrease by a constant multiplier. Sherwin, in his general discussion of such series, uses a and q; Chase, a and m, the former considering the series under the idea of a common quotient and the other under that of a common multiplier. In order to preserve the r in general discuscussions and have a short word, the common multiplier might be called the rate. Already has the term ratio as applied to equidifferent series been nearly supplanted by common difference. Although the French consider rapport and raison as conveying the same idea, they do not in practice speak of the rapport of a progression but of the raison.

"Le rapport de chaque terme au précédent renomme raison."— Briot.

"Ce nombre de fois est ce qu'on appelle la raison de la progression."— Bezout. "Ce rapport constant, qui existe entre un terme et celui qui le précède immediatement, se nomme la RAISON de la progression.”- - Bourdon.

I have not observed any departure from the use of raison for the rate of an equimultiple series in French works.

THE best bow should sometimes be unstrung.

THE TRUANT.

My teacher was a merry man,
And people called him witty;
He skilfully could "logic chop,"
Could preach or write a ditty.
My mother sent me off to school, -
Her rules were few but rigid,—
To learn of heat and polar snows,
Of temperate zones and frigid.

My task was light but ne'er performed, —
The time I idly wasted,—

My teacher soon his patience lost,

And soundly I was "basted."

I longed to leave the school-room's din,
And be with truants playing;

I thought that learning two times three,
Was harder far than haying.

The morning come, my class was called, —
The truth I now am telling,-

But I had packed my books and left,
Which pleased me more than spelling.
The teacher urged my quick return,
But loudly I protested;

My sisters joined their plea with his, -
These were alike detested.

I sought the camp and battle field
Renowned in ancient story;

I dreamed of wearing cap and plume,
And fighting all for glory.

'T was thus I lived full fifty years,
All learning proudly scorning;
I now am wretched, poor, despised,
From me let youth take warning.

THAT education is incomplete which develops only one side of our nature. We cannot unduly exercise one faculty, without neglecting others; thus left to themselves, they soon become weakened by disuse.