78 



♦ KNOAA^LEDGE ♦ 



[February 1, 1887. 



interested in the study of double stars and stellar systems 

 will no do\ibt be glad to learn," they add, " that photography 

 may hereafter be applied effectively in these important re- 

 searches." 



MINUTE MEASUREMENT. 



one in the least degree familiar with science 

 in its historical aspect can be ignorant of 

 the extent to which it has been indebted for 

 its enormous advancement to the increased 

 and increasing delicacy of physical measure- 

 ments. Nor is this merely true in connec- 

 tion with any single branch of natural 

 knowledge. Whether (taking a very few illustrations almost 

 at random) we select the instruments emploj'ed by Tycho 

 Brahe, for comparison with those now to be found in 

 every first-class observatory in the world ; the rude 

 balances of the alchemists with the exquisite masterpieces 

 of C)ertling ; the rough means of mechanical measurement 

 employed by Galileo, with the fittings of a modern physical 

 laboratory ; or the screws cut by Plumier at the beginning 

 of the last century with the mechanical marvels turned out 

 by Sir Joseph Whitworth, we shall alike be struck with 

 the manner in which our knowledge of those bi-anches of 

 science in whose pureuit they are respectively employed 

 have advanced 2)a7-i passu with the improvements in them. 

 It may then be neither uninteresting nor uninstructive if 

 we attempt to give, in a popular form, some account of the 

 manner in which ordinarily insensible quantities are made 

 manifest and easily measurable ; and to this end we pro- 

 pose to describe in a familiar style some of the devices and 

 instruments employed in the measurement of extremely 

 minute quantities. Such description may enable those who 

 have previously devoted but scant attention to the subject 

 to realise with more force the trustworthiness of the data 

 on which scientific theories are now founded, and to appre- 

 ciate better the confidence with which modern men of science 

 regard their results. 



Suppose, then, in the outset, that we wish to meiisure a 

 distance to the one-hundredth of an inch upon a given 

 sti-aight line, in what way shall we proceed to do sol 

 Scarcely by dividing a straight scale so finely, and using that 

 as our standard ; inasmuch as the strokes of the divisions, 

 and the intervals separating them would probably differ too 

 little in width to be trustworthy by ordinary vision. 

 Perhaps one-fiftieth of an inch is the smallest quantity 

 which can be fairly employed for such a purpose by a person 

 of average sight, and one half of this may be estimated. If, 

 though, we are merely to employ estimation, a very little 

 practice will enable us to take off quantities of this order 

 of minuteness with considerable accuracy from a scale 

 divided into inches and tenths, as in fig. 1, 



I I 



u 



Fig. 1. 



where s s' represents a sc;ile so divided. If now we lequlre 

 to take off 1^ inch from this scale, we, of course, simply place 

 one point of our compasses on s and the other on 5 in the 

 figure to get what we want at once. If, though, we wish to 

 obtain a length of, say l-'2'2 inch, then again placing one 

 point of our compasses on s, we extend the other to a, as 

 nearly as we can estimate 0"2 di^•ision beyond the second 



division of our second inch, and thus we obtain the length 

 required. In a similar way it will be seen that the distance 

 between s and 6^1-56 inch, and so on. But, after all, 

 guessing to a certain extent enters into such deter- 

 minations, and men of science cannot in the least degree 

 afford to be dependent on guesswork. Let ns, then, see 

 whether we can find any more rigidly accurate method of 

 measuring a quantity so small as that of which we have been 

 speaking. We do find such a one in what is called the 

 " Diagonal Scale," which is engraved on the boxwood or 

 ivory protractor in every case of mathematical instruments — 

 a scale which will not onlj' enable us to measure to the one- 

 hundredth of an inch, but to the one-hundredth of even a 

 quarter of an inch (or the four-hundredth of an inch) if 

 necessary. The principle on which it is constructed will be 

 evident from a little study of fig. 2. 



012 3466780 10 



12 3 



Fig. 2. 



4 6 6 7 8 9 10 



Here we see eleven equidistant parallel lines, the upper 

 one of which is diHded into two of the primary units 

 (inches) in our figure, and perpendicular lines drawn 

 through them. In practice, of course, the scale shown above 

 would lie extended towards the left and divided into inches, 

 as in the case of a 1 o o. It is, however, with the right- 

 hand one of these primary divisions that we are here more 

 particularly concerned. This we subdivide into ten equal 

 parts, both upon the upper and lower lines. We now draw 

 straight lines from the zero point above to subdivision 1 

 below, from 1 above to 2 below, and so on, until we come to 

 subdivision 9 above, which is joined to 10 below. Then 

 obviously, as all these diagonal lines are parallel and 

 equidistant at every point, while the first one coincides 

 with the zero point at o on the top line, it must depart 

 one tenth of a subdivision from the perpendicular o o in 

 the second, two-tenths in the third, three-tenths in the 

 fourth . . . and so on to ten-tenths, or one entii'e sub- 

 division on the bottom line. Then evidently a o ^ 1 inch 

 and al (still measuring on the top line) I'l inch; bb', 

 however, as will be easily seen, = 1 inch -|- one-tenth of 

 one-tenth of an inch, or -01 inch ; in other words, 6 6' = 

 1-01 inch. Similarly, cc' = l'02 inch. So again, J d' = 

 1 inch -f- "2 inch -f -04 inch — i.e. to 1'24 inch ; as does ee' 

 to 1"88 inch, and so or. for clearness, we have drawn our 

 figure to a one-inch scale, but on the ivory protractors in 

 cases of instruments a quarter of an inch scale will be found 

 to be divided in this way : from which, as we have said 

 above, ^^^Vjjtli of an inch should, theoretically, be susceptible 

 of measurement. Perhaps half this quantity, though, is all 

 that can be rigidly depended on with ordinary compasses, 

 But in describing the construction and use of the diagonal 

 scale, we have presupposed the employment of compasses, 

 or something analogous, to take off the quantity we require : 

 and it must be obvious, on the slightest reflection, that 

 no such mode of proceeding is, or can be, applicable in such 

 cases as those of the measurement of the height of the 

 mercury in a barometer, or angular deviation on the " limb," 

 or circular periphery, of a circle employed to measure angles 

 with. It is true that a diagonal scale Wixs engraved on the 

 limbs of astronomical quadrants by Cantzler in England 

 towards the end of the sixteenth century, and was adopted 



