Biometry

“ To realize what is meant when an angle is measured with a “probable error” of a few hundredths of a second of arc, it should be remembered that one second of arc on a circle 10 inches in diameter is less than 1/40000 of an inch. The accuracy which has been attained in the measurement of base lines is not easily realized by a layman. An engineer realizes the practical impossibility of measuring a line twice and obtaining356 precisely the same result to the finest unit of measurement. ”

“ But since these are the species of the whole mathematical science, geometry is again divided into the contemplation of planes, and the dimension of solids, which is called stereometry. For there is not any peculiar treatise about points and lines, because no figure can be produced from these without planes or solids. For geometry treats of nothing else in every one of its parts, than that it may constitute either planes or solids: or that when constituted, it may compare and divide them among themselves. ”

“ The best way to introduce this theorem is to cut a parallelogram from paper, and then, with the class, separate it into two parts by a cut perpendicular to the base. The two parts may then be fitted together to make a rectangle. In particular, if we cut off a triangle from one end and fit it on the other, we have the basis for the proof of the textbooks. The use of squared paper for such a proposition is not wise, since it makes the measurement appear to be merely an approximation. The cutting of the paper is in every way more satisfactory. ”

“ The statement that there is just one straight line between two points and that this is the shortest possible path between the points is entirely different in character from any information ever given by physical measurement, for all our measurements are subject to error. It is possible, nevertheless, to give a certain real physical meaning to the ideally precise statements of geometry, because it is a result of everyday experience that as we refine the accuracy of our physical measurements the quantitative statements of geometry are verified within an ever decreasing margin of error. ”

“ Even when we have defined a particular type of measurement, be it Euclidean or non-Euclidean, we have by no means fixed the behaviour of our rods and bodies in any absolute sense; and an understanding of absolute rigidity still escapes us. We can only define Euclidean bodies, for example, as those whose laws of disposition yield Euclidean numerical results. The following illustration will show us that these bodies might all vary in absolute shape when displaced, yet still yield the same Euclidean results. ”

“ Yet it is obvious that when they proceed to measure the earth’s diameter, travelling at the same speed, the two observers will reach the same figure for it.
Hence the π which expresses the proportion of the earth’s circumference to its diameter on the ground of actual measurement differs according as the measurer travels in the direction of the earth’s rotation or in the opposite direction. Therefore, as the real values of π are different, they cannot be the unique and quite definite figure of classical geometry. ”

“ For a plane surface, the ratio of the circumference of a circle to its diameter, both lengths being measured with the same rod, is, according to Euclidean geometry of the plane, equal to a constant value , which is independent of the diameter of the circle. On their spherical surface our flat beings would find for this ratio the value i.e. a smaller value than , the difference being the more considerable, the greater is the radius of the circle in comparison with the radius R of the “world-sphere.” ”

“ Through knowledge of the times one is able to determine other unknowns, as was seen in section 200. One who knows the measure of arc BC or angle BEG can determine, through use of the tables, how many units are needed to function as a secant for that angle, and after the measure of the maximum sine is determined, which is line BE, or half the diameter of the Earth, it can now be seen that the height of the mountain is the difference between the maximum sine and the secant line. ”

“ When numbers, however, are taken in a concrete sense, and are applied to express peculiar modes of quantity,—such as angular, linear, superficial, and solid magnitudes,—a new set of numerical relations arise, and a large number of computations are required.
To express angular magnitude, and the various relations of linear magnitude with which it is connected, involves the consideration of a vast variety of Geometrical and Trigonometrical tables; such as tables of the natural sines, co-sines, tangents, secants, co-tangents, &c. &c.; tables of arcs and angles in terms of the radius ”

“ Indeed, the late Professor Newcomb stated that "ten decimal places are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimal places would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope." ”

“ It is mathematically equal to the product of the length of the circumference, multiplied by half the radius. To square a circle of a given diameter in mètres, is the same as giving the number of squares, of a mètre in each side, of which the surface is the equivalent. If, the diameter being given, the exact circumference were known by a sort of inspiration, the superficial extent of the circular space would be deducible from the two numbers, by the mere multiplication of the numerical length of the circumference by the fourth of the diameter, or half the radius. ”

“ The question whether the practical geometry of the universe is Euclidean or not has a clear meaning, and its answer can only be furnished by experience. All linear measurement in physics is practical geometry in this sense, so too is geodetic and astronomical linear measurement, if we call to our help the law of experience that light is propagated in a straight line, and indeed in a straight line in the sense of practical geometry. ”

“ The answer is that the requirements of scientific measurement are growing more and more rigorous every year. A hundred years ago a measurement made to within one-thousandth of an inch was considered rather phenomenal. Now it is one of the modern requirements in the most accurate machine work. At present a few measurements are relied upon to within one ten-thousandth of an inch. There are cases in which an accuracy of one-millionth of an inch has been attained, and it is even possible to detect differences of one five-millionth of an inch. ”

“ Ordinary measurement by a foot rule, a surveyor’s chain, or the excessively careful measurements of the base line of a trigonometrical survey by standard bars, are sufficient instances of this procedure.
In the second case, where p y/x = q, we multiply or divide a magnitude until we get what is equal to the unit, or to some magnitude easily comparable with it. As a general rule the quantities which we desire to measure in physical science are too small rather than too great for easy determination, and the problem consists in multiplying them without introducing error. ”

“ We have seen above that we constantly reason as if the geometrical figures behaved like solids. What geometry would borrow from experiment would be therefore the properties of these bodies. The properties of light and its propagation in a straight line have also given rise to some of the propositions of geometry, and in particular to those of projective geometry, so that from that point of view one would be tempted to say that metrical geometry is the study of solids, and projective geometry that of light. ”

“ The trigonometrical functions are employed in many branches of mathematical and physical science not directly concerned with the measurement of angles, and hence arises the importance of analytical trigonometry. The solution of triangles of which the sides are geodesic lines on a spheroidal surface requires the introduction of other functions than those required for the solution of triangles on a plane or spherical surface, and therefore gives rise to a new branch of science, which is from analogy frequently called spheroidal trigonometry. ”

“ When the standard quantity is greater than that to be measured, we often adopt the first mode, and subdivide the unit until we get a magnitude equal to that measured. The angles observed in surveying, in astronomy, or in goniometry are usually smaller than a whole revolution, and the measuring circle is divided by the use of the screw and microscope, until we obtain an angle undistinguishable from that observed. The dimensions of minute objects are determined by subdividing the inch or centimetre, the screw micrometer being the most accurate means of subdivision. ”

“ But when a few base lines have been obtained in different places on the earth's surface, the measuring rods are to be laid aside, and the subsequent task of the survey of the earth is to be conducted by the measurement of angles from one station to another and trigonometrical calculations based thereon. Starting from a base line a few miles long, distances of greater length are calculated, until at length stretches 100 miles long, or even more, can be accomplished. ”

“ They therefore set to work to reconstruct geometry on a basis which did not assume the universal possibility of numerical measurement—a reconstruction which, as may be seen in Euclid, they effected with extraordinary skill and with great logical acumen. The moderns, under the influence of Cartesian geometry, have reasserted the universal possibility of numerical measurement, extending arithmetic, partly for that purpose, so as to include what are called “irrational” numbers, which give the ratios of incommensurable lengths. ”

“ This requires a new proof, as we must substitute for the words 'the bisector of the angle FAC,' the words 'the perpendicular to FC drawn through A.'
(5) Vertical angles together greater than two right angles.
This also requires a new proof, as we must insert, after the words 'the bisector of the angle FAC,' the words 'produced through A,' and must then prove (by your Th. 1) that the angles OAC, OAF, are equal.
On the whole, I take this to be the most cumbrous proof yet suggested for this Theorem. ”

“ If we now determine a line , such that this integral is equal to
,
where and are the areas of the two sections, the length of will be the same whatever unit of length we adopt, and whatever system of logarithms we use. If we suppose the sections divided into elements of equal size, then the logarithm of , multiplied by the number of pairs of elements, will be equal to the sum of the logarithms of the distances of all the pairs of elements. Here may be considered as the geometrical mean of all the distances between pairs of elements. ”

“ Persons are still to be met with who seem to have some illusions on this point, and who see in the doctrine of the dimensions of the units a doctrine of general physics, while it is, to say truth, only a doctrine of metrology. The knowledge of dimensions is valuable, since it allows us, for instance, to easily verify the homogeneity of a formula, but it can in no way give us any information on the actual nature of the quantity measured. ”

“ This is also one of the basal propositions of solid geometry, and it has many applications in practical mensuration. A first-class textbook will give a sufficient list of problems involving numerical measurement, to fix the law in mind. For outdoor work, involving measurements near the school or within the knowledge of the pupils, the following problem is a type:
If this represents the cross section of a railway embankment that is l feet long, h feet high, b feet wide at the bottom, and b ́ feet wide at the top, find the number of cubic feet in the embankment. ”

“ If we take the word measurement in its direct and general mathematical acceptation, which signifies simply the determination of the value of the ratios between any homogeneous magnitudes, we must consider, in geometry, that the measurement of surfaces and of volumes, unlike that of lines, is never conceived, even in the most simple and the most favourable cases, as being effected directly. ”

“ Besides which, the cubit being the unit, the half cubit might serve the purpose of the foot; but the metre, divisible only by two and ten, gave no measure practically corresponding to the foot whatever. It appears to have been considered that decimal arithmetic, although affording great facilities for the computation of numbers, is not equally well suited for the division of material substances. A glance of the eye is sufficient to divide material substances into successive halves, fourths, eighths, and sixteenths. A slight attention will give thirds, sixths, and twelfths. ”

“ A man ignorant of geometry may perhaps lament, that the circumference of a circle does not happen to be exactly three times the diameter, or in some other assignable proportion to it, rather than such that we have not yet been able to explain what the ratio between them is ”

“ Run a line from A to C by standing at C in line with A and P. Then run two perpendiculars from A and C by any of the methods already given,—sighting on a protractor or along the edge of a book if no better means are at hand. Then sight from some point D, on CD, to P, putting a stake at B. Then run the perpendicular BE. Since DE : EB = BA : AP, and since we can measure DE, EB, and BA with the tape, we can compute the distance AP.
There are many variations of this scheme of measuring distances by means of similar triangles, and pupils may be encouraged to try some of them. ”