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951
BINOMIAL

of Alfred Nachet, who in 1853, and subsequently in 1863, brought forward two forms of binocular microscope.


1911 Britannica-Binocular -Telestereoscope1.png

Fig. 5.


The earliest stages of the development of the binocular microscope had been always confined to those instruments with one objective, in the immediate neighbourhood of which the systems for dividing the pencil were placed. At a later date attempts were made to separate the two halves of the objective by modifying the eye-piece; this led to the construction of stereoscopic eye-pieces, initiated by R. B. Tolles, E. Abbe and A. Prazmowski. Of special importance is the work of Abbe; although, as he himself has stated, his methods accidentally led to the Wenham system, he certainly was far above his predecessors in his theoretical treatment of the problem, and in the perspicuity and clearness of his explanation. To him is also due the re-establishment of the instruments, which Wenham had abandoned by reason of too great technical difficulties (fig. 8). The newest form of the binocular microscope is very similar to the oldest form in which two completely separated tubes were employed. The inventor, H. S. Greenough, employs two systems for setting up the image, in order to avoid the pseudoscopic effect. After experiments in the Zeiss works, the erecting of Porro’s prisms simultaneously permitted a convenient adaptation to the eye-distance of the observer.


1911 Britannica-Binocular -Microscope.png
Fig. 6.Fig. 7.Fig. 8.


1911 Britannica-Binocular -Microscope1.png

Fig. 9.

The first binocular magnifying glass or simple microscope (German, Lupe) was devised by J. L. Riddell in 1853; in this instrument (fig. 9) the pencil of light is transmitted to the eyes by means of two pairs of parallel mirrors. Simple microscope. Of the many different improvements mention may be made of A. Nachet’s. H. Westien made use of two Chevalier-Brücke’s simple microscopes with their long working distances in order to form an instrument in which the curvature of the image was not entirely avoided. Mention may also be made of the binoculars of K. Fritzsch (formerly Prokesch) and E. Berger.

Binocular Instruments for Range-finding.—For measuring purposes binocular telescopes with parallel axes are the only types employed. The measurement is effected by adjoining to the space or interval to be measured some means of measurement defined; for example, by a fixed scale which extends into the space, or by a movable point (Wandermarke). This instrument shows a transition to the stereoscope, inasmuch as the scale or means of measurement is not directly observed, but to each eye a plane representation is offered, just as in the stereoscope; the space to be measured, on the other hand, is portrayed in exactly the same way as in the double telescope. The method for superposing the two spaces on one another was deduced by Sir David Brewster in 1856, but he does not appear to have dealt with the problem of range-finding. The problem was attacked in 1861 by A. Rollet; later, in 1866, E. Mach published a promising idea, and finally—independently of the researches of his predecessors—Hektor de Grousilliers, in partnership with the Zeiss firm (E. Abbe and C. Pulfrich), constructed the first stereoscopic range-finder suitable for practical use.  (O. Hr.) 

BINOMIAL (from the Lat. bi-, bis, twice, and nomen, a name or term), in mathematics, a word first introduced by Robert Recorde (1557) to denote a quantity composed of the sum or difference to two terms; as a + b, ab. The terms trinomial, quadrinomial, multinomial, &c., are applied to expressions composed similarly of three, four or many quantities.

The binomial theorem is a celebrated theorem, originally due to Sir Isaac Newton, by which any power of a binomial can be expressed as a series. In its modern form the theorem, which is true for all values of n, is written as

(x + a)n = xn + naxn−1 +   n·(n − 1)   a2xn−2   n·(n − 1)·(n − 2)   a3xn−3 ... + an.
1·2 1·2·3

The reader is referred to the article Algebra for the proof and applications of this theorem; here we shall only treat of the history of its discovery.

The original form of the theorem was first given in a letter, dated the 13th of June 1676, from Sir Isaac Newton to Henry Oldenburg for communication to Wilhelm G. Leibnitz, although Newton had discovered it some years previously. Newton there states that

(p + pq)m/n = pm/n +   m   aq +   mn   bq +   m − 2n   cq ... &c.,
n 2n 3n

where p + pq is the quantity whose mth/n power or root is required, p the first term of that quantity, and q the quotient of the rest divided by p, m/n the power, which may be a positive or negative integer or a fraction, and a, b, c, &c., the several terms in order, e.g.

a = pm/n, b =   m   aq, c =   mn   bq, and so on.
n 2n

In a second letter, dated the 24th of October 1676, to Oldenburg, Newton gave the train of reasoning by which he devised the theorem.

“In the beginning of my mathematical studies, when I was perusing the works of the celebrated Dr Wallis, and considering the series by the interpolation of which he exhibits the area of the circle and hyperbola (for instance, in this series of curves whose common base or axis is x, and the ordinates respectively (1 − xx)0/2, (1 − xx)1/2, (1 − xx)2/2, (1 − xx)3/2, &c.), I perceived that if the areas of the alternate curves, which are x, x1/3x3, x2/3x3 + 1/5x5, x3/3x3 + 3/5x51/7x7, &c., could be interpolated, we should obtain the areas of the intermediate ones, the first of which (1 − xx)1/2 is the area of the circle. Now in order to [do] this, it appeared that in all the series the first term was x; that the second terms 0/3x3, 1/3x3, 2/3x3, &c., were in arithmetical progression; and consequently that the first two terms of all the series to be interpolated would be x1/2x3/3, x3/2x3/3, x5/2x3/3, &c.

“Now for the interpolation of the rest, I considered that the denominators 1, 3, 5, &c., were in arithmetical progression; and that therefore only the numerical coefficients of the numerators were to be investigated. But these in the alternate areas, which are given, were the same with the figures of which the several powers of 11 consist, viz., of 110, 111, 112, 113, that is, the first 1; the second, 1, 1; the third, 1, 2, 1,; the fourth 1, 3, 3, 1; and so on. I enquired therefore how, in these series, the rest of the terms may be derived from the first two being given; and I found that by putting m for the second figure or term, the rest should be produced by the continued multiplication of the terms of this series m − 0/1 × m − 1/2 × m − 2/3 ..., &c. ... This rule I therefore applied to the series to be interpolated. And since, in the series for the circle, the second term was (1/2x3)/3, I put m = 1/2.... And hence I found the required area of the circular segment to be x1/2x3/31/8x5/51/16x7/7, &c. ... And in the same manner might be produced the interpolated areas of other curves; as also the area of the hyperbola and the other alternates in this series (1 + xx)0/2, (1 + xx)1/2, (1 + xx)2/2, &c. ... Having proceeded so far, I considered that the terms (1 − xx)0/2, (1 − xx)2/2, (1 − xx)4/2, (1 − xx)6/2, &c., that is 1, 1 − x2, 1 − 2x2 + x4, 1 − 3x2 + 3x4x6, &c., might be interpolated in the same manner as the areas generated by them, and for this, nothing more was required than to omit the denominators 1, 3, 5, 7, &c., in the terms expressing the areas; that is, the coefficients of the terms of the quantity to be interpolated (1 − xx)1/2 or (1 − xx)32, or generally (1 − xx)m will