THE 
METRICAL SYSTEM 
OF 
WEIGHTS AND MEASURES. 
BY 
H. A. NEWTON, 
PROFESSOR OF MATHEMATICS IN YALE COLLEGE. 
NEW HAVEN: 
PRINTED BY E. HAYES, 426 CHAPEL ST. 
1864. The following pages have been written to facilitate the teach- 
ing, in common schools, of a system of weights and measures 
which is in partial or exclusive use in France, Italy, Spain, Austria, 
the larger part of Germany, Holland, Belgium, Portugal, Greece, 
and other States. It is in extensive use in science, is employed 
to some extent in this country in certain departments of the arts, 
and its terms are becoming common even in our popular litera- 
ture. A bill permitting and providing for its use in connection 
with the present weights and measures has passed the House of 
Commons, has passed to a third reading in the House of Lords, 
and will doubtless become a law of the British realm. A new 
nomenclature will probably be adopted by the Parliament. A 
committee of the House of Representatives has under considera- 
tion its introduction into the U. S. Custom houses and Mints. It 
is not unlikely that it will be thus adopted, with or without modi- 
fications, at some not very remote day, in which case the system 
will be sooner or later legalized by the several State legislatures. 
The Statistical Congress assembled at Berlin last year, composed 
of delegates from nearly all the Governments of the Christian 
world, recommended that the system be taught in the public 
schools. This action -was laid before the authorities of the State 
of Connecticut by Hon. S. B. Ruggles, the delegate to that Con- 
gress from the U. S. Government, whereupon the Legislature 
adopted a resolution recommending to the proper school officers 
to provide for such instruction in the schools under their charge. 
These pages may serve as a supplement to any arithmetic. 
By several authors, tables of the Metrical system are given, but 
they are rarely, if ever, accompanied by problems, and the sub- 
ject is not afterwards referred to. These books can hardly be 
said to teach the system. 
I have taken for the length of the metre 39'3685 IT. S. standard 
inches, that being the length used in the IT. S. Coast Survey. The 
gallon is assumed to contain 231, and the bushel 2150'42 cubic 
inches. The weight of the kilogram as given by Capt. Kater, 
that is, 2'20606 avoirdupois pounds, is used. 
It is to be hoped that an incorporation of the subject into the 
arithmetics will, at an early day, render this supplementary tract 
superfluous. h. a. n. 
Yale College, August 30th, 1864. THE METRICAL SYSTEM 
OF 
WEIGHTS AND MEASURES. 
The metre is the basis of the system, and is the principal unit for meas- 
uring lengths. It was intended to be, and is very nearly, one ten-millionth 
of the distance, on the earth’s surface, from the equator to the pole, or one 
forty-millionth of the circumference of the globe measured over the poles. 
It is equal to 39-3685 standard inches. 
The are is a square whose side is ten metres. It is the principal unit for 
superficial measure. 
The litre is a cube whose edge is the tenth part of a metre. It is the 
principal unit for all measures of capacity. It is equal to 0-26414 wine gal- 
lons, or a little more than a wine quart. 
The stere is a cube whose edge is a metre. It is the unit for wood meas- 
ure. The stere is 1000 litres. 
The gram is the unit of weight, and is the weight of a cube of pure 
water, each edge of the cube being -j-g-jth of a metre. It is equal to about 
15-44 grains. 
The water is to be weighed in a vacuum, and is to be at the temperature of greatest 
density, which is about 5° Centigrade, or 39° Fahrenheit. 
The franc is the principal unit of value. It is a coin weighing five 
grams, and is nine-tenths silver and one-tenth copper. 
In France bronze coins are made representing one, two, five, and ten, hundredths of 
a franc. These weigh severally one, two, five, and ten grams. 
Each of these several units is divided into ten parts, each of these parts 
into ten smaller parts, and again each of these into ten still smaller parts. 
On the other hand, multiples of the principal units by 10,100, 1000, &c., form 
larger units. 
To designate multiples by 10, 100, 1000, and 10,000, the prefixes deca, 
hecto, kilo, and myria, are severally applied. To designate the 10th, 100th, 
and 1000th parts the prefixes deci, centi, and milli, are given. 
The former series was derived from the Greek language, the latter from the Latin. 4 
One myriametre = 10 kilometres, = 893685 inches, 
one kilometre — 10 hectometres, = 89368'5 “ 
one hectometre — 10 decametres, = 3936'85 “ 
one decametre = 10 metres, = 393'685 “ 
one metre = 10 decimetres, = 39'3685 “ 
one decimetre = 10 centimetres, = 3'93685 “ 
one centimetre — 10 millimetres, = 0'393685 “ 
Measures of Length. 
Measures of Capacity. 
One kilolitre (or stere) = 10 hectolitres = 100 decalitres = 1000 litres 
= 10,000 decilitres = 100,000 centilitres = 1,000,000 millilitres. 
Weights. 
One tonneau, or millier = 10 quintals = 100 myriagrams = 1000 kilograms 
= 10,000 hectograms = 100,000 decagrams =1,000,000 grams =10,000,000 
decigrams = 100,000,000 centigrams = 1,000,000,000 milligrams. 
The kilogram (usually called kilo) is the ordinary weight of commerce, 
and is'equal to 2'20606 pounds avoirdupois. 
The same prefixes apply, in theory, to the are and the stere. But of the 
terms which are thus formed, only the hectare, the centiare (square metre), 
the decastere, the decistere, and the centistere, are in common use. The only 
terms in use which refer to the franc are the decime and centime, which ex- 
press 10th and 100th parts. 
Like Federal money, these weights and measures are all written decimally, 
and operations upon them involve only the common rules of arithmetic. Thus 
8 kilometres, 1 decametres, 5 metres, 1 centimetre, and 8 millimetres are writ- 
ten 8015'018 metres, or 8'015018 kilometres. To reduce one denomination to 
a higher or lower denomination, remove the decimal point one or more places 
to the right or left, adding or prefixing cyphers if necessary. 
Examples. 
1. What is the value per litre of wine that costs 250 francs a hectolitre ? 
Ans. 2'5 francs. 
2. What is the value of 15 kilograms at $500 a tonneau? Ans. $1.50. 
8. What is the value of 28 grams at $150 a kilogram ? Ans. $4.20. 
4. The value of a kilogram being 432'25 francs, what is the value of a 
gram? Ans. '43225 of a franc. 
5. What is the value of 3 metres 1 decimetres and 5 centimetres of cloth, 
at 4 francs 48 centimes per metre ? Ans. 4'48 X 315 = 16'8 francs. 
6. Two hectolitres and 16 litres of wine cost 128 francs; what must it be 
sold for per litre to gain 20 per cent on the cost ? Ans. 11 -f* centimes. 5 
7. If a man walks 50 kilometres a day, in how many days could ho walk a 
distance equal to the meridional circumference of the earth ? Ans. 800. 
8. A floor is 3 metres and 25 centimetres long, and 6 metres, 48 centimetres 
broad; what is the surface ? 
Ans. 3 25 X 6’48 = 21 06 centiares or sq. metres. 
9. A board is 1 decimetre and 27 millimetres broad, and 5 metres and 21 
centimetres long. How many square decimetres in the board ? 
Ans. 1-27 X52-1 = 66 167. 
10. How many metres of carpeting 84 centimetres broad would cover a 
surface of 27 square metres? Ans. 3214-f- metres. 
11. A cistern 4 metres square, and two metres deep, can contain how many 
hectolitres of water? Ans. 320. 
12. A box is 2 metres long, 15 decimetres broad, and a metre deep; how 
many hectolitres of grain will it contain ? Ans. 30. 
13. A stone is 2 metres long, 75 centimetres broad, and 24 centimetres 
thick; what is its volume ? A ns. 0’42 cubic metres. 
14. Find the value of a stone 2 metres, 3 decimetres and 4 centimetres 
long. 1 metre, 4 decimetres and 7 centimetres wide, and one metre, 75 centi- 
metres thick, at 22 francs a stere, or cubic metre ? 
Ans. 22 X 234 X 1’47 X1‘75 francs. 
15. What is the weight of a litre of water? Ans. 1 kilogram. 
16. What is the weight of a cubic metre of water? Ans. 1 tonneau. 
17. A cistern is 3 metres long, 2 metres wide, and one metre deep; what 
weight of water will it contain ? Ans. 6000 kilograms, or 6 tonneaux. 
Substances are often compared as to weight with the weight of the same 
bulk of water. If a body of a given size weighs a certain number of times 
as much as the same bulk of water, that number is called the specific gravity 
of the body. Thus if a cubic inch of iron weighs as much as 7| cubic inches 
of water, its specific gravity is 7'5. Again, if the weight of a cubic inch of 
cork is one-fourth that of a cubic inch of water, the specific gravity of the 
cork is 0’25. 
Gases are compared with air instead of water. 
The following table shows the specific gravity of several substances. As 
different specimens of the same material may differ greatly in density, the 
numbers must be regarded as only approximately correct. 
Weights of Bodies. 
Copper 8’8 
Gold 19-3 
Iron (cast) 7'2 
Iron (bar) 7’8 
Silver (pure) 10’4 
Lead 1T4 
Mercury 13'5 
Elm 0-7 
Mahogany T1 
Maple 0'8 
Granite 2’7 
Marble 2’7 
To find then the weight of a body, multiply the weight of an equal volume 
of water by the specific gravity. It should be remembered that a litre of 
water weighs a kilogram, and a cubic metre weighs a tonneau. 6 
Examples. 
1. What is the weight of a cubic metre of cast iron? Ans. 7'2 tonneaux. 
2. What is the weight of two litres of mercury? Ans. 27 kilos. 
3. What is the weight of a cubic metre of marble? Ans. 2'7 tonneaux. 
4. What is the weight of a granite block 1 metre long, 32 centimetres 
thick, and 75 centimetres broad? 648 kilos. 
5. What is the weight of a maple board 3 metres long, 5 decimetres wide, 
and 2 centimetres thick? Ans. 24 kilos. 
6. What is the weight of a bar of iron 4 metres long, 1 decimetre broad, 
and 3 centimetres thick? Ans. 93 6 kilos. 
7. The weight of a hectolitre of wheat is about 75 kilograms; what weight 
of wheat would fill a bin 2 metres long, T4 metres broad, and 1 metre deep? 
Ans. 2100 kilos. 
Comparison with the U. S. Standard Weights and Measures. 
The following table gives the multiplier or divisor for reducing metres, li- 
tres, Ac., into feet, inches, quarts, Ac. In order to reduce the common weights 
and measures into grams, metres, Ac., multiply by the divisor, or divide by 
the multiplier. 
The multipliers and divisors are given in most cases to four or five decimals 
only. Greater accuracy may be obtained in comparing lengths, surfaces, and 
capacities, by considering the metre as 39-3685 U. S. inches, and in comparing 
weights by considering the kilogram as 2'20606 pounds avoirdupois. The 
gallon is called 231, and the bushel 2150'42 cubic inches. 
Multiply metres by 39-3685 or divide by -0254 to get inches. 
“ metres “ 3'2807 “ -30481 “ feet. 
“ metres “ 109357 “ -91444 “ yards. 
“ metres “ -19883 “ 5 0294 “ rods. 
“ kilometres “ -62135 “ 1-6094 “ miles. 
“ sq. metres “ 1550- “ -0006452 “ sq. inches. 
“ sq. metres “ 10 763 “ -09291 “ sq. feet. 
“ sq. metres “ 1196 “ -8362 “ sq. yards. 
“ ares 3-953 “ ' -2529 “ sq. rods. 
“ hectares “ 2-4709 “ -4047 “ acres. 
“ hectares “ -003861 “ 259- “ sq. miles. 
“ litres “ 33-81 “ -02958 “ fluid ounces, 
litres “ T05656 “ -9465 “ quarts. 
“ litres “ -26414 “ 3-7S6 “ gallons. 
“ hectolitres “ 2-837 “ -3524 “ bushels. 
“ litres “ 61-012 “ -01639 “ cub. inches. 
“ hectolitres “ 3-531 “ -2832 “ cub. feet. 
“ steres “ 1-3078 “ -7646 “ cub. yards. 
“ steres “ -2759 “ 3'625 “ cords. 
“ grams “ 1544 “ -0648 “ grains. 
“ kilograms “ 32-147 “ -03108 “ Troy ounces. 
“ kilograms “ 35-30 “ -02833 “ Avoir, ounces. 
“ kilograms “ 2 681 “ -373 “ Troy pounds. 
“ kilograms “ 2-206 “ -4536 “ Avoir, pounds. 
“ tonneaux “ -985 “ 1-015 “ long tons. 
“ tonneaux “ 1103 “ -9066 “ short tons. 7 
1. Reduce 125 metres to yards. Ans. 125 X 1’09357 = 136'696 -f- 
2. Reduce 25 hectolitres to bushels. 25 X 2'837 =70 925 bush. 
3. Reduce 25 gallons to litres. Ans. 25 X 3'786 =94’65 litres. 
4. Reduce 160 kilograms to pounds avoirdupois. 
5. Reduce 17 kilometres to miles. 
6. Reduce 150 acres to hectares. 
7. Reduce 3 oz. Troy to grams. 
8. Reduce 12 feet 6 inches to millimetres. 
9. Reduce 150 bushels to litres. 
10. In 127 miles how many decimetres? 
11. In a township 6 miles square how many hectares? 
12. In 15 cords of wood how many steres? 
13. In 187 pounds avoirdupois how many kilos ? 
14. Reduce 1 kilogram to pounds, ounces, <fcc., in apothecary’s weight. 
15. Reduce 10 fluid ounces to millilitres. 
16. Reduce 12'5 grains to milligrams. 
Examples. 
Approximate values of the units of the two systems. 
Rude approximations are for many purposes quite sufficient. In such cases, 
ratios for conversion that are easily remembered are useful. The following 
are suggested. 
1 decimetre = about 4 inches. 
1 millimetre = “ Jjth of an inch. 
5 metres = “ 1 rod. 
1 kilometre = “ 200 rods, or |ths of a mile. 
1 hectare = “ 2£ acres. 
1 litre = “ 1-j quarts. 
1 hectolitre = “ 2f bushels, or of a barrel. 
1 kilogram = “ 21 pounds avoirdupois. 
tonneaux= “ 1 long ton. 
The metre is about three feet, three inches, and three eighths of an inch. 
Conversion, of Prices. 
When the price per unit in one system is given, the price per unit of the 
other system may be found by the multipliers and divisors of the table on 
the preceding page, since the prices are as the magnitudes of the units. We 
may use the following rules. 
Multiply the price for any unit on the left hand side of the table by the 
corresponding divisor, and the product is the price for the unit on the right 
hand side. 
Multiply the price for any unit on the right by the first number in the same 
line, and the product is the price for the unit at the left. 8 
Examples. 
1. The price per pound avoirdupois being 75 cents, what is it per kilogram! 
Ans. -75 X 2-206 = $l-65-f-. 
2. The price per litre being 62 cts., what is it per quart ? 
Ans. 62 X ‘9465 = 59 cts. nearly. 
3. The produce per hectare being 7 hectolitres, what is it per acre ? 
Ans. 1 X '4047 = 2 83 4~ hectolitres. 
4. The price per gallon being 12 francs, what is it per litre? 
Ans. 12 X "26414 = 3*17—francs. 
5. The produce per square mile being 7600 bushels, what is it per hectare? 
Miscellaneous Examples. 
1. How many pounds avoirdupois in a bar of iron 2 metres long, 3 deci- 
metres broad, and 15 millimetres thick? 
2. What is the value of a cubic metre of cast iron at $45 a short ton ? 
3. What is the value of a cubic metre of silver, of the purity of coin, the 
specific gravity being called 10 ? Ans. 2,000,000 francs. 
4. What is the value of a sheet of copper 2 metres square and 5 millime- 
tres thick, at 50 cts. a pound ? 
5. If 15 kilos cost $165, what is the price per ounce troy ? 
6. If 17 yards cost $16.20, what is the price per metre ? 
7. If 215 litres cost 150 francs, what must be the price per gallon to gain 
25 per cent ? 
8. In 3 pounds 6 ounces 2 scruples and 17 grains, how many grams ? 
9. Reduce 1725 grams to apothecaries weight? 
10. What is the weight in kilos of 1000 cubic feet of maple timber ?