Art. XI.—A System of Weights and Measures.
[Read before the Wellington Philosophical Society, 1st September, 1877.]
In designing a system of weights and measures there are several points to be taken into consideration, of which the most important is, perhaps, that the radix of the system shall be continually divisible by two without a remainder. The number of inches, for instance, which the foot contains or the number of shillings which the pound contains should be some power of two. An odd number would be most inconvenient as the radix. If the foot contained eleven inches, half a foot would contain 5.½ inches, and a fraction is at once introduced, the inconvenience of which in commerce and in all arithmetical and mechanical work is very great.
Next to the odd numbers, the most inconvenient are the odd numbers multiplied by two such as 6, 10, 14, 18, etc. Here the objectionable fraction is put off one step only, and on halving twice again shows itself.
No system of measures in which one of these numbers is adopted as the radix ever has been nor ever will be thoroughly in use. The American divides his dollar into half and quarter dollars, and to continue as far as he can the convenience of being able to divide by two he adopts the “bit” or “York shilling” unknown to the law. The English workman divides the inch, not into three barley-corns as by law directed, but into halves, quarters, eighths, and sixteenths. The French workman again divides his millimetre into halves and quarters like his English brethren. The two systems, in fact, run side by side but do not coalesce; as far as the decimal system lends itself to division by two it is used, but no further. As soon as it fails in this respect it is thrown aside in favour of the more natural and convenient system of having a radix continually divisible by two.
Another important point is that the several measures of weight, superficies, capacity, etc., shall be tied together, and be interdependent. The French adopted a logical system in which this point received full attention, but the inherent unsuitability of the number 10 as a radix has prevented its adoption in full, and their system is now a body without a head, for their
unit of length, although it is a determinate part of the circumference of the earth, bears no simple proportion to a degree of latitude. The English system is a complete chaos, but in fixing the imperial gallon a slight movement in this direction was made, as it was fixed as a vessel which would just contain ten pounds weight of water.
The next point for consideration is that the units shall be of convenient value. All the European nations have adopted a measure of length not very different from the English foot. This may be taken, therefore, as a measure which has been proved by experience to be sufficiently convenient. Considerable latitude is, however, permissible in this matter, and no inconvenience has followed the adoption of the metre which is more than three times as long as the English foot. Whatever value be adopted for the unit it should be as far as possible dependent on some measure given by nature and not liable to change.
The last point and one of the most important of all is, that the radix of measures shall be the same as the radix of counting. This latter all over the world is the number 10. Unfortunately this number is not well suited for the radix of measures, and as long as it is maintained as the radix of counting, we cannot have a perfect system of weights and measures.
We cannot alter the properties of the number 10, but we can depose it from its undeserved eminence as our counting radix, and adopt a more suitable number. If our primitive forefathers had only turned in their thumbs, when using their hands to help them in the difficult process of counting, and had used their eight fingers instead of their ten fingers and thumbs, we should have had 8 for our counting radix—a nearly perfect number, as it is a power of two. They did not do so; but there is no reason why we should be bound by their mistake for ever. We should clearly change the faulty radix we have inherited from them.
The difficulty of changing the counting radix has, however, always been looked upon as too great to attempt, and the scientific world is quietly adopting the French system, knowing it to be imperfect and inconvenient, although less so than any other in use, having nearly all the requirements of a perfect system, except that of having for its radix some power of two. This exception is, however, of sufficient importance to condemn the system. The inherent unsuitability of ten as a radix will become more and more apparent, as commerce, and the arts and sciences take a more important place in daily work. It must sooner or later be set aside for some other number. We are, therefore, only putting off the evil day by adopting the French system, and would much better change the radix at once.
I cannot but think that the difficulty of making the change is very much exaggerated. We have some experience of the ease with which a radix
other than 10 is adopted, as in some of the more trifling affairs of business the dozen has actually supplanted the ten. For instance, eggs are sold by the dozen, and 12 has become the counting radix as far as eggs are concerned. Everyone feels at once the number represented by six dozen eggs; but if 72 eggs are mentioned, the number is instinctively turned into dozens before a clear perception of its value is obtained. If, now, everything were sold by the dozen, and the pound contained a dozen shillings, and all other weights and measures were reckoned in dozens, it would become a very simple thing to count by the gross and dozen instead of by the hundred and the ten. Twelve fails in the necessary requirement that it shall be continually divisible by two without a remainder, and is not therefore suitable for the radix; but, if 16 or 8 were used as a radix of measures, it would after a few months, or at most a few years, be so easy to reckon in sixteens or eights, instead of in tens, and at the same time so convenient, that legislation would not be required to effect the change; it would come of itself by use and habit, just as it has come about that eggs are now reckoned in dozens. Some slight inconvenience might be felt at first, such as is felt in going to a foreign country, where new measures are met with; but everyone who has lived abroad knows how slight this inconvenience is. After a few months the new measures quite supplant the old; and it becomes convenient, in thinking of English measures, to translate them into those which even so short an experience has rendered familiar. It is, in fact, only through pure cowardice to meet a difficulty that the scientific world is taking up the imperfect French system. A far better system, which would never require alteration, might be adopted, if we would only fairly look in the face the difficulty of changing the counting radix; and, like most difficulties, this seems the less the more it is looked at.
Assuming then that the radix must be changed the question arises what number is to supplant ten. It must be a power of two. Two itself and four are too small. Eight has some claims but is also too small. School-boys would all vote for it as they would have to learn the multiplication table only up to 8 times 8 instead of to 10 times 10 as at present, but the inconveniences of having so small a radix are too great and a larger must be sought. Thirty-two on the other hand is too large. The average mathematical mind would not be able to work a multiplication table extending to 32 times 32, and a loss of convenience would accrue. Half-way between these two would be about right, that is 16 should be the radix. The multiplication table would not be unwieldily large and the figures required to express a large number would not be too numerous. The present radix 10 is certainly smaller than is desirable, and 16 would be an improvement from every point of view.
Our radix being thus established we must next establish our unit of length on which all the others depend. The earth is our best natural measure, and our standard sea-mile should represent a definite angle measured on its surface, as our present sea-mile or knot does. All commercial nations, including the French, have been driven to adopt this measure for the purposes of navigation, although in no case does it correspond with any measure used on the land. The increasing importance of commerce makes it very desirable that the land-mile and the sea-mile should be the same. This brings us to angular measurement as the foundation of the whole system.
The French saw the absurdity of dividing the circle into ten equal parts, so they divided it into four right angles, and then divided each right angle into ten parts. On this division of the earth's circumference they founded their measures of length. Unfortunately the circle is not to be influenced by Acts of Parliament or of Senate. It is by its very nature divided not only into four right angles but also into six sections of equal importance to the right angle, and any system of angular measure which ignores this fact must break down. The French system ignored it so completely that the important angle of 60° cannot even be expressed in figures. It consequently broke down completely; it never had the least chance of coming into use and is now seldom heard of.
It is essential that the division of the circle shall be such that the right angle, and the arc of which the chord is equal to radius, shall both be expressed by convenient numbers. If the arc of 60° be divided into 16 equal parts, 24 of such parts would represent the right angle, and this would, with the radix of 16, be the best division possible. On this system the earth's circumference would be divided into 6 equal parts, which we may call radius arcs; each of these would be divided into 16 points, each point into 16 degrees, and each degree into 16 sea-miles. The circle would be divided in the same manner, and the sea-mile would represent a measure of latitude corresponding to a second of the present measures. The new seamile would be a little longer than the present statute mile, and somewhat shorter than the present sea-mile or knot; it would be 5,350 feet long, while the statute mile contains 5,280 feet. Of course the mile would be further sub-divided, always into sixteens, which would give us the ell equal to about 16 inches; and the inch, which would differ only about 2 per cent. from the inch as we now have it.
Measures of weight follow from those of length. A cubic ell of water would weigh 137lbs., and would be the standard. Larger or smaller measures would be obtained by multiplying or dividing by 16, as might be required.
The cubic ell and superficial ell would be the fundamental measures of capacity and area.
Measurement of time is only a form of circular measurement, and the day should be divided, like the circle, into six watches of four hours each; the watch would be divided into sixteen parts, each exactly equal to a quarter of an hour; and the quarter-hour into sixteen minutes, instead of fifteen as at present.
I need not further recite the different measures to be used, as I affix a table showing their value in ordinary English measures.
Of course fifteen new figures would have to be designed to use with the sixteen-fold system of counting, as the present figures would have to be kept exclusively for the decimal system. In the tables I have used the letters of the alphabet instead of the new figures.
The system I have sketched out would have all the advantages of the decimal system and none of its disadvantages. It would be coherent throughout, and would greatly reduce the labour required in all arithmetical processes arising in business and science. There would be a loss of money in making the change, as a large amount of capital has been invested in machinery which has been designed for sub-dividing the inch in England, and the corresponding measure in other countries. As far as England, Russia, and America are concerned, this might be saved by a small sacrifice of the completeness of the system. By taking the inch as the standard, and multiplying by sixteen for the higher measures, we should get a mile which would be about 2 per cent. shorter than the sea-mile.
[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]
|Decimal System||Sixteen-fold System|
|1||1||Day||1 Day equal 24 hours|
|6||6||Watch||1 Watch " " 4 " "|
|96||96||" "||1 " " " " ¼ " "|
|1536||1536||" "||1 " " " " 1/1.5/6 minute|
|1||1||Circle||1 Circle equal 360 degrees|
|6||6||Radius arcs||1 Radius are " " 60 " "|
|96||96||" "||1 " " " "4 " "|
|1596||1596||" "||1 " "" " 15 minutes|
|4,096||4,096||" "||1 " "" " 15/16 " "|
|1||1||Sea-mile||1 Sea mile equal to 5,350 feet|
|16||16||" "||1 " " 335 " "|
|256||256||" "||1 " " 21 " "|
|4,096||4,096||Ells||1 Ell " " 1.3 " "|
|65,536||65,536||" "||1 " " " " 0.98 inch|
|1,048,576||1,048,576||" "||1 " " " " 1/16 " "|
|1||1||Square mile||1 Square mile equal 688 acres|
|16||16||" "||1 " " 43 " "|
|256||256||" "||1 " " 2.2/3 " "|
|4,096||4,096||" "||1 " " ⅙ " "|
|65,536||65,536||" "||1 " " 453 sq. feet|
|1,048,576||1,048,576||" "||1 " " 28 " "|
|16,777,216||16,777,216||Square ells||1 Square ell " " 1.8 " "|
|1||1||Cube||1 Cube equal 35 cubic feet|
|16||16||Cubic ells||1 Cubic ell " " 2.2 " "|
|256||256||New gallons||1 New gallon " " ⅞ gallon|
|4,096||4,096||" "||1 " " " " ⅕ pint|
|1||1||Load||1 Load equal 2,197 lbs.|
|16||16||Water ells||1 Water ell " " 137 " "|
|256||256||" "||1 " " " " 8.6 " "|
|4,096||4,096||" "||1 " " " " ½ " "|
|65,536||65,536||Water inch||1 Water inch " " ½ oz.|
Hunt.—Notes on blowing up Snags in the Waikato River with Dynamite. 161