215. When the Science of Astronomy was much less perfect than it is at present, the length of the solar year was much less accurately known; and accordingly we find that in the time of Julius Cæsar, it was supposed to consist of 365 days, 6 hours, or of 365 days, exactly. On this supposition, it is evident that if, out of four years in succession, any three consisted of 365 days each, and the remaining one of 366 days, the Sun would have returned, at the end of these four years, to the place in the ecliptic which it occupied at their commence

ment.

The scheme was therefore called the Julian Calendar; and if the hypothesis had been correct, it would have been attended with much convenience: the additional day was made by repeating the Sixth of the Calends of March in the Roman Calendar, which corresponds with the 24th of February in ours: also, the year in which it was inserted was termed Bissextile, and the additional day was called Intercalary, on that account.

This regulation, applied to the years of the Christian Era, was so conducted that, whenever the number of years was divisible by 4, the corresponding year consisted of 366 days, and was called Leap-year, the month of February having 29 days in that year, and in each of the remaining three years only 28 days, without interfering at all with their order.

Hence also, the remainder after the division of any other number of years by 4, was the number of years since a leap year occurred up to that year: thus, in the year 1839 this remainder is 3; and accordingly it is 3 years since the last leap year happened, and it is 1 year before the next will occur, according to this scheme.

216. Since the true solar year is 365.242264 days, and not 365.25 days, it is evident that the reckoning of time, according to the Julian Calendar, would place the end of the year after the time when the Sun had returned to the point of the ecliptic occupied by it at the beginning of the year, and consequently in advance of the course of the seasons:

but, the error in one year is

365.25-365.242264 = .007736 of a day:

whence, by finding how often this is contained in 1 day,

will be the number of years in which the error amounts to 1 day also, by the rule of proportion,

129.2657 yrs. 400 yrs. :: 1day. : 3.0944 days, nearly: whence, it follows that 3.0944 days, or 3 days, 2 hours, 16 minutes, is very nearly the error which would accumulate in 400 years.

Now, according to the Julian Calendar, 400 years would comprise 100 leap years; and since we find that this reckoning falls nearly 3 days after the true time, it is evident that if there were only 97 leap years in the same space of time, the year corresponding would very nearly agree with the true solar year; and it is accordingly ordained that whenever the numbers expressing the Centuries, as 16, 17, 18, 19, &c., denoting 1600, 1700, 1800, 1900, &c., are not divisible by 4, the corresponding year shall not be a leap year, although according to the Julian Computation it would: thus, 1600 would be a leap year, but 1700, 1800, 1900, would not.

The calendar thus corrected, though not absolutely accurate, is very well adapted to every practical purpose, as the error in 5000 years will not amount to much more than twenty-eight hours. This correction was first promulgated in Europe by Pope Gregory in the year 1582, and the calendar has since been called the Gregorian Calendar, but it was not introduced into Protestant Countries till a much later period. In England, it was adopted on the second day of September 1752, when the error amounted to 11 days: and it is called the New Style, to distinguish it from the Julian Calendar, which is now termed the Old Style.

Had the old style continued, the error would now have been 12 days, because 1800 would, according to it, have been a leap year, which in the new it was not: and thus, we have in Almanacks, Old Christmas-day, Old Midsummer-day, &c., taking place 12 days after the times in which they are fixed by our present system.

Though all the calculations of modern times are conducted by means of the new style, a knowledge of

the difference of the two styles is not without its use, both in the perusal of old Documents, and in the Astronomical Verification of Historical Facts, which could not be performed without it.

217. The common civil year thus fixed and determined, is then subdivided into twelve Calendar Months, as described in the Table. The word Month however, is frequently used in different senses: sometimes to denote a twelfth part of the year or 30 days; sometimes as equivalent to 4 weeks or 28 days; and accordingly, a year is said to be equivalent to 13 months and 1 day, or to 52 weeks and 1 day, with the addition of another day when it happens to be leap year.

FRENCH IMPERIAL MEASURES, &c.

218. In consequence of the irregularity in the measures and multiples of all the units just mentioned, it is evident that the calculation of measures and weights will be much more complicated and difficult, particularly to Foreigners, than if they were connected by some common divisor and multiplier; and it was with the view of obviating this inconvenience, that a New System of measures and weights has been adopted in France.

219. In this system, the length of the Terrestrial Arc from the Equator to the Pole in the Meridian of Paris, is taken as the General Standard; and the following Synopsis of French Measures exhibits them as compared with the standard of this country.

We may here observe that the Metre, or one ten millionth part of the Terrestrial Arc, is the Element of lineal measure; the Are or Square Decametre, that of superficial measure; the Stere or Solid Metre, that of solid measure, and the Litre or Cubic Decimetre, that of the measure of capacity.

220. In like manner the Weights belonging to this system, and expressed in English grains, are

Here the Gramme is the Element, being the weight of a cubic centimetre of distilled water.

221. The Angular Measures in the same system expressed in English Degrees, are as follow:

Here 100 Grades are consequently equivalent to 90 Degrees in the English scale: and in the inferior denominations, the Centesimal scale is uniformly used by the French, where the English proceed according to the Sexagesimal.

222.

The unit of Value in France is a silver coin called a Franc, consisting of ths of pure silver and th of alloy and its subdivisions are as follow:

The value of an English pound sterling is equivalent to that of 25.2 francs, very nearly: and thus, the value of 1 franc expressed in English money is

223. Wherever this system is used, it is evident that the Theory of Decimals, as laid down in the fifth Chapter, will be sufficient for performing all the fundamental operations of Arithmetic, entirely superseding what has been done in the second chapter of this work.

The Student, who may be desirous of prosecuting his enquiries in this very interesting and important subject, is referred to the Articles, Weights and Measures, in BARLOW'S Mathematical and Philosophical Dictionary, and to the last edition of DR. KELLY'S Universal Cambist.

PROBLEMS.

224. We will conclude the Application of Arithmetic to Geometry, with the consideration of a few Problems of common occurrence, in the solution of which, the principles explained in this chapter are generally taken for granted.