Weighing by a Ternary Series of Weights

Abstract

IT has been shown in NATURE (vol. xlii. p. 568) that any number of pounds may be weighed with weights, the numbers of pounds in which form a geometrical progression with 1 for first term and 3 for common ratio. The following method of treating the same problem may serve to illustrate some remarks made by the President of the Mathematical Section of the British Association at the recent meeting in Leeds. One of these remarks had reference to the fascinating interest attaching to such inquiries into the properties of series of numbers, another showed that the adoption of special systems of notation for different problems was often of great service, and a third remark alluded to the attainment of one and the same result by diverse methods of procedure. In the present case the interest attaching to the subject may be left to speak for itself; the notation suitable for the problem requires elucidation. It is well known that by means of only two figures, 1 and 0, any number may be expressed if we agree that the value of the 1 shall be doubled every time it is removed one place further to the left, so that, for example, 11111 would denote the number 1 + 2 + 4 + 8 + 16, and that any number not greater than 31 would be denoted by means of five figures or less. It follows that if we had five weights of corresponding value to the above five numbers we could weigh any number of units of weight from 1 to 31. Now, the present problem only differs from this in two respects—namely, in that the 1 increases threefold in value on being removed one place to the left, and that the value denoted by it may in any position, except the place on the extreme left, be taken negatively. Let us agree to denote the negative value by using a different type, and we may then indicate all values from 1 to 40 as follows:—