6. 364 myrialitres, 47 litres, 384 millilitres. 7. 243 decalitres, 47 centilitres. 8. 9. Ans. L 364004.7384. Ans. 1L 243.047. Unit of Number, Second Order of Multiples. 23 myriametres, 72 millimetres. Ans. 2M 2300.00072. 4000007 steres and 2 millisteres. Ans. 2S 40000.07002. RULE III. When a submultiple of a principal unit of measure is the unit of number; - First, Place before the number the initial letter of the principal unit from which the submultiple is derived. Second, Indicate the order of submultiple used by a small figure placed to the left and below the letter prefixed to the number. (See symbols in table of submultiples.) EXAMPLES FOR PRACTICE. Write the numbers which represent the following quantities, com sidering the denomination named as the unit of number. Ans. 2S 302005003. 5. 5 kilogrammes and 9 grammes. 6. 302 myriasteres, 5 decasteres, and 3 centisteres. 7. 4009 kilolitres and 5 litres. 8. 2 hectares and 2 centiares. Ans. L 400900500. Ans. 2A 20002. RULE FOR REDUCTION DESCENDING. Multiply the given quantity by the number of the required denomination which makes a unit of the given denomination. Since the multiplier is always 10, 100, 1000, &c., the operation is performed by removing the decimal point as many places to the right as there are ciphers in the multiplier, annexing ciphers when by the number of its own denomination which makes a unit of the required denomination. Since the divisor is always 10, 100, 1000, &c., the operation is performed by removing the decimal point as many places to the left as there are ciphers in the divisor, prefixing ciphers when necessary. RELATIONS OF UNITS OF SURFACE TO UNITS OF LENGTH. Decimilliare = One square decimetre = 100 Milliare Centiare = {10 = square centimetres. plane figure whose one decimetre. 10 square decimetres, or a { :{" 10 square metres, or a plane figure whose length is one decametre and breadth one metre. One square decametre 100 square metres. 10 square decametres, or a plane figure whose length 10 square hectometres, or a plane figure whose length One square kilometre 100 square hectometres. NUMERAL EXPRESSION FOR SURFACE. The contents of a plane figure is expressed numerically by giving the number of times it contains some given area, which is assumed as the unit of surface. The following illustrations will show how the various denominations of the table are used in numerical expressions of surface :— ILLUSTRATION FIRST. Breadth 3 metres. Length 6 metres. It will be seen, by examining this figure, that the lines drawn parallel to the sides, at the supposed distance of a metre from each other, divide the surface into square metres, and that there are as many rows of square metres as there are metres in the breadth, each row containing as many square metres as there are metres in the length. IIence the number of square metres in the area of the figure is equal to the product of the two numbers which indicate the length and breadth; and A 0.21 is a numerical expression for its contents. Breadth 1 decametre, 2 metres, and 1 decimetre. 6 decimetres. ILLUSTRATION SECOND. 6 metres. In this figure, the lines drawn parallel to the sides divide the figure into 36 milliares, or oblongs, whose length is one metre and breadth one decimetre. It is evident that ten of these oblongs put together will constitute a centiare, or square metre. Hence the expression, 36 milliares, may be written 3.6 centiares; and read, three and six tenths centiares, or three centiares and six milliares. By reducing the length to decimetres, the numerical expression of the contents will be, by Illustration First, 60 x 6, or 360 decimilliares or square decimetres. ILLUSTRATION THIRD. Length 1 decametre, 2 metres, and 1 decimetre. Milliare. Decimilliare. In this figure, we have illustrated the relations of different denomi nations of units in expressing the contents of a given surface. In the following analysis, each part of the contents is presented separately, as it would be obtained by multiplying the length by the breadth. The learner should carefully note each part, and analyze a sufficient number of examples to fix the principles in the mind. From these illustrations, we derive the following rule for finding a numerical expression for a given surface of uniform length and breadth : RULE. Reduce the length and breadth to the same denomination, find the product of the two dimensions after reduction, and point off as many decimal places in this product as there are decimal places in the two dimensions. The unit of the numerical expression thus found will be a decimilliare when the unit of length is a decimetre, a centiare when the unit of length is a metre, an are when the unit of length is a decametre, a hectare when the unit of length is a hectometre, and a myriare when the unit of length is a kilometre. EXAMPLES FOR PRACTICE. 1. How many ares in a floor M 1.25 long, and M 8.7 wide? Ans. A .10875. 2. How many centiares, how many kilares, and how many hectares in the same floor? Ans. A 10.875. 3. How many ares in a board M 5.32 by M 47.? Ans. A .025004. 4. How many milliares, how many myriares, and hectares in the same board? 5. How many metres of a carpet nine decimetres wide will cover |