The following table exhibits this system of nomenclature, and the advantages to be derived from a uniformity in the scale of notations: TABLE OF THE FRENCH DECIMAL MEASURES. EXAMPLES OF THE COMPOUND NAMES TO EXPRESS DIFFERENT UNITIES OF MEASURES. Myriamètre, a length of ten thousand mètres. Kilogramme, a weight of one thousand grammes. Hectare, an agrarian measure of one hundred ares. Décalitre, a measure of capacity, of ten litres. Décimètre, the tenth part of a mètre. Centigramme, the hundredth part of one gramme. The unity of money is called a Franc. The Franc is divided into ten Décimes, and the Décime into ten Centimes. TABLE OF COMPARISON BETWEEN THE FRENCH AND ENGLISH WEIGHTS AND MEASURES. A Mètre 3.2808992 feet, or 39-3707 inches. A Square Mètre 10-764297 square feet. A Litre = 2200967 gallons imperial, or 1·7607736 pints. A Gramme = 15.4378 grains. An Are024711 acres, or 119.60124 square yards. SUGGESTIONS RELATIVE TO A NEW SYSTEM OF WEIGHTS AND MEASURES FOR THIS COUNTRY. We should have only three measures, viz., a measure of length, a measure of capacity, and a measure of weight. Our present distinctions of dry and liquid measure, of Troy and avoirdupois weights, &c., should be abolished; the measure for beer should belong to the same scale as the measure for corn or potatoes, &c.; and all our measures should be graduated according to a decimal scale. MEASURE OF LENGTH AND VOLUME. The first thing to be done is to fix the unit of length; upon this unit the units of capacity as well as of weight must be based. I propose that our present foot should be adopted as the unit of length. All our most simple conceptions of distance and length are associated with the foot measure. Artificers of all kinds have their foot rules by their sides, and mark off all their sizes by it; the draper has his yard wand, or three feet measure; and the builder has his tape line graduated into units of feet. To change the foot, therefore, would not only disturb all our existing associations relative to length, size, and distance, but would introduce confusion into our present commercial transactions. The foot is very nearly the 32,809,000th part of the distance between the pole and the equator, or it is the 3.2659th part of the length of a second's pendulum at London; so that, if the government standard of a foot should be lost, it might be readily restored. The French mètre is the ten-millionth part of the distance from the pole to the equator, but the advantage that would be derived from this decimal scale of derivation would not, in my opinion, compensate for the evils that would arise from the alteration of our present unit of measure. Taking the foot as our unit of length, the new scale of lengths would be as follows: The foot divided into ten equal parts would give us a TENTH, or a DÉCIFOOT, which might be taken as the new inch. A The foot divided into one hundred equal parts, or the new inch into ten equal parts, would give us a HUNDREDTH or a CENTIFOOT; and so on. Ten feet would give us a TEN-FEET MEASURE, or a DÉCAFOOT. hundred feet would give us a HUNDRED-FEET MEASURE, or a HECTOFOOT, which might be taken as the new chain. A thousand feet would give us a THOUSAND-Feet measure, or a KILOFOOT; and so on. A foot square gives us a unit of surface, which we might call an area; and a foot cube gives us a unit of volume, which we might call a cube. : MEASURE OF CAPACITY. Having fixed the unit of length, the unit of capacity should be derived from the unity of length by some simple law of ratio. Of all our existing measures of capacity we seem to be most familiar with the pint. Now it happens that the volume of our half-pint is exactly the hundredth part of a cubic foot for 16 half-pints, or our present imperial gallon, contain 10 pounds of distilled water, that is, a half-pint weighs 10 oz. ; now a cubic foot of distilled water weighs 1,000 oz. ; therefore the capacity of a cubic foot is 100 times that of a half-pint. Hence the HALFPINT might be called a CENTICUBE, and the PINT might be called Two CENTICUBES. According to this scale our new measures of capacity would be as follow: The MILLICUBE, or the thousandth part of a cubic foot, or the tenth part of our half-pint. The CENTICUBE, or the hundredth part of a cubic foot, or our present half-pint. The DÉCICUBE, or the tenth part of a cubic foot, or ten half-pints, which might be called the new pottle, or new half-gallon. The CUBE, or a cubic foot, or one hundred half-pints, which might be called the new bushel. The DÉCACUBE, or ten cubic feet, or one thousand half-pints, which might be called the new quarter, or the new hogshead. The present quarter contains 1,024 half-pints, and the present hogshead of wine 1,008. The HECTOCUBE, or one hundred cubic feet, or ten thousand half-pints, which might be called the new last; the present last contains 10,240 half-pints. MEASURE OF WEIGHT. The unit of weight, like the unit of capacity, should be derived from the unit of length by some simple law. There is no weight with which we are more familiar than the ounce. Now it happens that the ounce avoirdupois is exactly the thousandth part of the weight of a cubic foot of distilled water at a mean temperature, or 62°. This therefore affords us the means of recovering the standard ounce if it should be lost. Taking the ounce, therefore, as the standard of weight, our new measures of weights might be written as follow: The CENTIDRAM, or the thousandth part of an ounce. This might be called the new grain. The DÉCIDRAM, or the hundredth part of an ounce. The NEW-DRAM, or the tenth part of an ounce. The DÉCADRAM, or the present ounce. The HECTODRAM, or ten ounces. This might be called the new lb. SCALES OF DEGREES. When the change is made in our weights and measures, it is also desirable that our present scales of degrees should be altered. Nothing can be more arbitrary and inconvenient than our scales of degrees, especially the scale of degrees used in our thermometers. The French scale of degrees is both uniform and philosophical. According to this scale, the quadrant of the circle is divided into one hundred equal parts, and each part is called a degree of the circle, or an angular degree. The freezing point of water on the thermometer is called zero, and the boiling point is called 100 degrees, so that between these points there are 100 equal divisions, each of which measures a degree of temperature. T. T. SOLUTION OF ARITHMETICAL PUZZLES. TO THE EDITOR OF THE ENGLISH JOURNAL OF EDUCATION. SIR,-I send you some proofs of curious properties of numbers, both according to the Binomial Theorem, and in a simpler form, so as to suit the capacity of some of the pupils, whom you may wish to explain this matter to. There is an 8vo. volume-"Barlow's Theory of Number". -which contains many curious properties of numbers equally striking e. g. the sum of the digits, in the odd places of any number, be equal to the sum of the digits in the even places, the number is divisible by 11. Thus, take 1 5 6 4 3 5 2 2 4. Here, 1+6+3+2+4=16: and 5+4+5+2=16. And the quotient, when divided by 11, is 14221384. This may be shown by a similar kind of proof. The above number =4+2×10 +2.102+5.103+3.101=4+2. (11−1)+2. (11 −1)2+5 . (11-1)3, &c., and, expanding by the Binomial Theorem=4+2× (11-1) +2. (112−2 × 11+1)+5.(113—3 × 112+3.11−1, &c., every term of which will contain some multiple of 11, except the last, and that will be +1, or −1, alternately. Therefore, the number=11n. +4-2+2-5 +3-4+6-5+1. If, therefore, the positive and negative numbers exactly counterpoise each other, the number is divisible by 11. SIMPLER PROOF. Let the digits of the number be a, b, c, d, e, f,-a being in the place of units, b of tens, &c. then, the number= a+10b+100c+1000d+10000e+100000ƒ =9 9+1×6+99+1.c+999+1.d+(9999+1).e+(99999+1)ƒ =a+b+c+d+e+f+9b+99c+999d+9999e +99999f. Now, 9.6+99.c+999.d, &c., is clearly divisible by 9, whatever the digits b, c, d, &c., be. If therefore, a+b+c, &c., as the sum of the digits, be divisible by 9, the number itself is.-Q. E. D. COR. I. The number=a+b+c+d+9b+99c, &c. ..Taking away the sum of the digits a+b+c, &c., the number =96+99c, &c., which is clearly divisible by 9. COR. II. If the digits of the first number be a, b, c, d, e, and ƒ, the number whose digits are the same, but in a reverse order, will be f, e, d, c, b, a. Or the first number= a+b+c+d+e+f+9b+99c+999d+9999e +99999ƒ. And the second number= f+e+d+c+b+a+9e +99d+999c+99996 +99999a. .. The difference between the two numbers= 99999+9990e+900d-900c-99906-99999a. Every term of which is divisible by 9; so, therefore, the number itself is divisible by 9. Example.-1798542, is a number whose sum of digits is divisible by 9; and therefore the number itself is. The sum of the digits=36, which is divisible by 9. The number itself, divided by 9=199838. The number 123456789, multiplied by 9, is the same thing as if it were multiplied by 10-1; As, first multiplying by 10 1234567890 the number will manifestly have 1 in every place except the second; so, if the numbers, 12345, be taken, this multiplied by 9, will have 1 in every place except the two last, or 111105. For, multiplied by 10, we get 123450 12345 111105 This will obviously be an analogous result, with respect to any of the numbers consisting of any numbers of the digits, 1, 2, 3, &c., in regular order, except that which ends in 9, and the only reason for a difference in that case, is because 9 × 9 is the only simple multiple of 9, which has 1 in the unit's place. PROOF BY BINOMIAL THEOREM. A number is divisible by 9, when the sum of the digits is divisible by 9. Let these digits be a, b, c.... .r, s, t, taken from right to left. =a+9+1.6+9+12. c, &c.,+9+1-3.r+9+1"-2.8+9+1n-1.t. Now if any of these binomials, as 9+1"-3, be expanded by the Binomial Theorem, the factor 9 will enter into every term of the expansion except the last, and the last term will always be 1. expanded term will, in fact be 9-3+n-3×9n-4-3 × 4.9n-5, &c., +n−3.9+1=93+1. The number will therefore be of the form The a+1.9+1.b+1.9+1.c, &c., +n±3.9+1.r+n22.9+1.8+n21.9+1.t =a+b+c+···+r+s+t+1.96+1.9c, &c., +39r, &c. Let 1.b+c++n=3.r+n=-3.8+1.t=F. Then the number=a+b+c+··· +r+s+t+9F. But 9F is clearly divisible by 97, if ::a+b+c···+r+s+t− be divisible by 9, the whole number is.—Q. E. D. COR. I. If from any number, the sum of its digits be subtracted, the remainder is divisible by 9. ..... +9 F. For the number=a+b+c, &c. Subtracting therefore a+b+c, &c., the remainder is 9F, which is clearly divisible by 9. COR. II. If from any number be subtracted that number whose digits are the same, but in a reverse order, the remainder will be divisible by 9. For the first number=a+b+c. ... +r+s+t+9. F. But the second number=t+s+r··· +c+b+a+9.. The difference between the two numbers=9F-9.4, which is clearly divisible by 9. H. |