or the mechanical equivalent of one C.G.S. Centigrade unit of heat

= 4'14 × 107 ergs.

If the agreement between scientific men as to the selection of fundamental units had been universal, a great deal of arithmetical calculation which is now necessary would have been avoided. There is some hope that in future one uniform system may be adopted, but even then it will be necessary for the student to be familiar with the methods of changing from one system to another in order to be able to avail himself of the results already published. To form a basis of calculation, tables showing the equivalents of the different fundamental units for the measurement of the same quantity are necessary. Want of space prevents our giving them here; we refer instead to Nos. 9-12 of the tables by Mr. S. Lupton (Macmillan & Co.). We take this opportunity of mentioning that we shall refer to the same work whenever we have occasion to notice the necessity for a table of constants for use in the experiments described.

1

CHAPTER III.

PHYSICAL ARITHMETIC

Approximate Measurements.

ONE of the first lessons which is learned by an experimenter making measurements on scientific methods is that the number obtained as a result is not a perfectly exact expression of the quantity measured, but represents it only withir

1 Numerical Tables and Constants in Elementa y Science, by S. Lupton.

certain limits of error. If the distance between two towns be given as fifteen miles, we do not understand that the distance has been measured and found to be exactly fifteen miles, without any yards, feet, inches, or fractions of an inch, but that the distance is nearer to fifteen miles than it is to sixteen or fourteen. If we wished to state the distance more accurately we should have to begin by defining two points, one in each town-marks, for instance, on the doorsteps of the respective parish churches-between which the distance had been taken, and we should also have to specify the route taken, and so on. To determine the distance with the greatest possible accuracy would be to go through the laborious process of measuring a base line, a rough idea of which is given in § 5. We might then, perhaps, obtain the distance to the nearest inch and still be uncertain whether there should not be a fraction of an inch more or less, and if so, what fraction it should be. If the number is expressed in the decimal notation, the increase in the accuracy of measurement is shewn by filling up more decimal places. Thus, if we set down the mechanical equivalent of heat at 4'2 x 107 ergs, it is not because the figures in the decimal places beyond the 2 are all zero, but because we do not know what their values really are, or it may be, for the purpose for which we are using the value, it is immaterial what they are. It is known, as a matter of fact, that a more accurate value is 4'214 × 107, but at present no one has been able to determine what figure should be put in the decimal place after the second 4.

Errors and Corrections.

The determination of an additional figure in a number representing the magnitude of a physical quantity generally involves a very great increase in the care and labour which must be bestowed on the determination. To obtain some idea of the reason for this, let us take, as an example, the case of determining the mass of a body of about 100

grammes. By an ordinary commercial balance the mass of a body can be easily and rapidly determined to 1 gramme, say 103 grammes. With a better arranged balance we may shew that 103 25 is a more accurate representation of the mass. We may then use a very sensitive chemical balance which shews a difference of mass of o'r mgm., but which requires a good deal of time and care in its use, and get a value 103 2537 grammes as the mass. But, if now we make another similar determination with another balance, or even with the same balance, at a different time, we may find the result is not the same, but, say, 103 2546 grammes. We have thus, by the sensitive balance, carried the measurement two decimal places further, but have got from two observations two different results, and have, therefore, to decide whether either of these represents the mass of the body, and, if so, which. Experience has shewn that some, at any rate, of the difference may be due to the balance not being in adjustment, and another part to the fact that the body is weighed in air and not in vacuo. The observed weighings may contain errors due to these causes. The effects of these causes on the weighings can be calculated when the ratio of the lengths of the arms and other facts about the balance have been determined, and when the state of the air as to pressure, temperature, and moisture is known (see §§ 13, 14).

We may thus, by a series of auxiliary observations, determine a correction to the observed weighing corresponding to each known possible error. When the observations are thus corrected they will probably be very much closer. Suppose them to be 103.2543 and 103 2542.

Mean of Observations.

When all precautions have been taken, and all known errors corrected, there may still be some difference between different observations which can only arise from causes beyond the knowledge and control of the observer. We

must, therefore, distinguish between errors due to known causes, which can be allowed for as corrections, or eliminated by repeating the observations under different conditions, and errors due to unknown causes, which are called 'accidental' errors. Thus, in the instance quoted, we know of no reason for taking 103 2543 as the mass of the body in preference to 103 2542. It is usual in such cases to take the arithmetic mean of the two observations, i.e. the number obtained by adding the two values together, and dividing by 2, as the nearest approximation to the true value.

Similarly if any number, n, of observations be taken, each one of which has been corrected for constant errors, and is, therefore, so far as the observer can tell, as worthy of confidence as any of the others, the arithmetic mean of the values is taken as that most nearly representing the true value of the quantity. Thus, if 91, 92, 93.... 9n be the results of the n observations, the value of q is taken to be

It is fair to suppose that, if we take a sufficient number of observations, some of them give results that are too large, others again results that are too small; and thus, by taking the mean of the observations as the true value, we approach more nearly than we can be sure of doing by adopting any single one of the observations.

We have already mentioned that allowance must be made by means of a suitable correction for each constant error, that is for each known error whose effect upon the result may be calculated or eliminated by some suitable arrangement. It is, of course, possible that the observer may have overlooked some source of constant error which will affect the final result. This must be very carefully guarded against, for taking the mean of a number of obser

D

vations affords, in general, no assistance in the elimination of an error of that kind.

The difference between the mean value and one of the observations is generally known technically as the 'error' of that observation. The theory of probabilities has been applied to the discussion of errors of observations', and it has been shewn that by taking the mean of n observations instead of a single observation, the so-called 'probable error' is reduced in the ratio of 1/Ö.

On this account alone it would be advisable to take several observations of each quantity measured in a physical experiment. By doing so, moreover, we not only get a result which is probably more accurate, but we find out to what extent the observations differ from each other, and thus obtain valuable information as to the degree of accuracy of which the method of observation is capable. Thus we have, on p. 72, four observations of a length, viz.—

Taking the mean we are justified in assuming that the true length is accurately represented by 3'333 to the third decimal place, and we see that the different observations differ only by two units at most in that place.

In performing the arithmetic for finding the mean of a number of observations, it is only necessary to add those columns in which differences occur-the last column of the example given above. Performing the addition on the other columns would be simply multiplying by 4, by which number we should have subsequently to divide.

An example will make this clear.

'See Airy's tract on the Theory of Errors of Observations.