Diagram of Developing Diagram of Developing Eye (1st stage). Eye (2nd stage). [alpha], Forebrain. [beta], Optic cup. [beta], Optic vesicle. [delta], Invagination of lens. [gamma], Superficial ectoderm. Other letters as in fig. 4.] [delta], Thickening for lens. Entry: FIG
and thus the motion in space of any point fixed in the body defined by [Lambda] is determined completely by means of [alpha], ß, [gamma], [delta]; and in the case of the symmetrical top these functions are elliptic transcendants, to which Klein has given the name of _multiplicative elliptic functions_; and Entry: 18
_Theorem._--If A, B, C, D are _Theorem._--If [alpha], ß, any four points in space, and if [gamma], [delta] are four planes the lines AB and CD meet, then in space, and if the lines all four points lie in a plane, [alpha]ß and [gamma][delta] meet, hence also AC and BD, as well then all four planes lie in a as AD and BC, meet. point (pencil), hence also [alpha][gamma] and ß[delta], as well as [alpha][delta] and ß[gamma], meet. Entry: 42
33. Let y be given as a function of x, or, more generally, let x and y be given as functions of a variable t. The first of these cases is included in the second by putting x = t. If certain conditions are satisfied the aggregate of the points determined by the functional relations form a curve. The first condition is that the aggregate of the values of t to which values of x and y correspond must be continuous, or, in other words, that these values must consist of all real numbers, or of all those real numbers which lie between assigned extreme numbers. When this condition is satisfied the points are "ordered," and their order is determined by the order of the numbers t, supposed to be arranged in order of increasing or decreasing magnitude; also there are two senses of description of the curve, according as t is taken to increase or to diminish. The second condition is that the aggregate of the points which are determined by the functional relations must be "continuous." This condition means that, if any point P determined by a value of t is taken, and any distance [delta], however small, is chosen, it is possible to find two points Q, Q´ of the aggregate which are such that (i.) P is between Q and Q´, (ii.) if R, R´ are any points between Q and Q´ the distance RR´ is less than [delta]. The meaning of the word "between" in this statement is fixed by the ordering of the points. Sometimes additional conditions are imposed upon the functional relations before they are regarded as defining a curve. An aggregate of points which satisfies the two conditions stated above is sometimes called a "Jordan curve." It by no means follows that every curve of this kind has a tangent. In order that the curve may have a tangent at P it is necessary that, if any angle [alpha], however small, is specified, a distance [delta] can be found such that when P is between Q and Q´, and PQ and PQ´ are less than [delta], the angle RPR´ is less than [alpha] for all pairs of points R, R´ which are between P and Q, or between P and Q´ (fig. 8). When this condition is satisfied y is a function of x which has a differential coefficient. The only way of finding out whether this condition is satisfied or not is to attempt to form the differential coefficient. If the quotient of differences [Delta]y/[Delta]x has a limit when [Delta]x tends to zero, y is a differentiable function of x, and the limit in question is the differential coefficient. The derived function, or differential coefficient, of a function [f](x) is always defined by the formula Entry: 33
If the glasses be in contact, as is usually supposed in the theory of Newton's rings, [alpha] = 0, and [delta]x[oo][lambda]
In older works Doric is often divided into a _dialectus severior_ and a _dialectus mitis_. But the difference is one of time rather than of place, the peculiarities of Doric being gradually softened down till it was ultimately merged in the _lingua franca_, the [Greek: koinê], which in time engulfed all the local dialects except the descendant of Spartan, Tzakonian. Here it is possible to mention its varieties only in the briefest form. (a) The southern dialects are well illustrated in the inscriptions of Laconia recently much increased in number by the excavations of the British School at Athens. Apart from some brief dedications, the earliest inscription of importance is the list of names placed on a bronze column soon after 479 B.C. to commemorate the tribes which had repulsed the Persians. The column, originally at Delphi, is now at Constantinople. The most striking features of the dialect are the retention of [digamma] at the beginning of words, as in the dedication from the 6th century [Greek: wanaxibios] (_Annual of British School_, xiv. 144). The dialect changed -[sigma]- between vowels into -h-, [Greek: môha] for [Greek: môsa] "muse." Later it changed [theta] into a sound like the English _th_, which was represented by [sigma]. Before o-sounds [epsilon] here and in some other Doric dialects changed to [iota]: [Greek: thios, sios] for [Greek: theos] "god." The result of contraction and "compensatory lengthening" was not [Greek: ei] and [Greek: ou] as in Attic and Ionic, but [eta] and [omega]: [Greek: êmen] infinitive = [Greek: einai] from *esmen; gen. sing. of _o_-stems in [omega]: [Greek: theô], acc. pl. in -[Greek: ôs: theôs]; dy was represented by [Greek: dd], not [zeta], as in Attic-Ionic; [Greek: musidde = muthize]. The dialect has many strange words, especially in connexion with the state education and organization of the boys and young men. The Heraclean tables from a Laconian colony in S. Italy have curious forms in -[Greek: assi] for the dat. pl. of the participle [Greek: prassontassi] = Attic [Greek: prattousi]. Of the dialect of Messenia we know little, the long inscription about mysteries from Andania being only about 100 B.C. From Argolis there are a considerable number of early inscriptions, and in a later form of the dialect the cures recorded at the temple of Asklepios at Epidaurus present many points of interest. There is also an inscription of the 6th century B.C. from the temple of Aphaia in Aegina. [Digamma] survives in the old inscriptions: [Greek: wewremena (= eirêmena); ns], whether original or arising by sound change from -_nty_, persists till the 2nd century B.C.: [Greek: hantitychonsa = hê antitychousa, tons huions = tous huious]. The dialect of the Inachus valley seems to resemble Laconian more closely than does that of the rest of the Argolic area. Corinth and her colonies in the earliest inscriptions preserve [Digamma] and [qoppa] (= Latin Q) before [omicron] and [upsilon] sounds, and write [xi] and [psi] by [Greek: chs] and [Greek: phs], the symbols which are used also for this purpose in old Attic. In the Corcyrean and Sicilian forms of the dialect, [lambda] before a dental appears as [nu]: [Greek: Phintias = Philtias]; and in Sicilian the perfect-active was treated as a present: [Greek: dedoikô] for [Greek: dedoika], &c. From Megara has come lately an obscure inscription from the beginning of the 5th century; its colony Selinus has inscriptions from the middle of the same century; the inscriptions from Byzantium and its other Pontic colonies date only from Hellenistic times. In Crete, which shows a considerable variety of subdialects, the most important document is the great inscription from Gortyn containing twelve tables of family law, which was discovered in 1884. The local alphabet has no separate symbols for [chi] and [phi], and these sounds are therefore written with [kappa] and [pi]. As in Argive the combination -[Greek: ns] was kept both medially and finally except before words beginning with a consonant; -_ty_- was represented by [zeta], later by -[Greek: tt]-, as in Thessalian and Boeotian: [Greek: hopottoi], Attic [Greek: hoposoi]; and finally by -[Greek: tt]-; [lambda] combined with a preceding vowel into an au-diphthong: [Greek: auka], Attic [Greek: alkê], cp. the English pronunciation of _talk_, &c. In Gortyn and some other towns -[Greek: st]--was assimilated to--[Greek: tt], where [theta] must have been a spirant like the English _th_ in _thin_; [zeta] of Attic Greek is represented initially by [delta], medially by [Greek: dd], but in some towns by [tau] and [Greek: tt: doos (= zôos), dikadden (= dikazein)]. Final consonants are generally assimilated to the beginning of the next word. In inflection there are many local peculiarities. In Melos and Thera some very old inscriptions have been found written in an alphabet without symbols for [phi], [chi], [phi], [xi], which are therefore written as [pi]h, [kappa]h or [koppa]h, [Greek: ps, ks]. The contractions of [epsilon] + [epsilon] and of [omicron] + [omicron] are represented by E and O respectively. The old rock inscriptions of Thera are among the most archaic yet discovered. The most characteristic feature of Rhodian Doric is the infinitive in -[Greek: mein: domein], &c. (= Attic [Greek: dounai]), which passed also to Gela and Agrigentum. The inscriptions from Cos are numerous, but too late to represent the earliest form of the dialect. Entry: 4
_Construction of Magic Squares._--A square of 5 (fig. 3) has adjoining it one of the eight equal squares by which any square may be conceived to be surrounded, each of which has two sides resting on adjoining squares, while four have sides resting on the surrounded square, and four meet it only at its four angles. 1, 2, 3 are placed along the path of a knight in chess; 4, along the same path, would fall in a cell of the outer square, and is placed instead in the corresponding cell of the original square; 5 then falls within the square. a, b, c, d are placed diagonally in the square; but e enters the outer square, and is removed thence to the same cell of the square it had left. [alpha], [beta], [gamma], [delta], [epsilon] pursue another regular course; and the diagram shows how that course is recorded in the square they have twice left. Whichever of the eight surrounding squares may be entered, the corresponding cell of the central square is taken instead. The 1, 2, 3, ..., a, b, c, ..., [alpha], [beta], [gamma], ... are said to lie in "paths." Entry: FIG
Both concepts have been elaborated and superseded by the modern procedure in respect to the axioms of geometry, and by the conception of abstract geometry involved therein. Riemann proceeds to specialize the manifold by considerations as to measurement. If measurement is to be possible, some magnitude, we saw, must be independent of position; let us consider manifolds in which lengths of lines are such magnitudes, so that every line is measurable by every other. The coordinates of a point being x1, x2, ... x_n, let us confine ourselves to lines along which the ratios dx1 : dx2 : ... : dx_n alter continuously. Let us also assume that the element of length, ds, is unchanged (to the first order) when all its points undergo the same infinitesimal motion. Then if all the increments dx be altered in the same ratio, ds is also altered in this ratio. Hence ds is a homogeneous function of the first degree of the increments dx. Moreover, ds must be unchanged when all the dx change sign. The simplest possible case is, therefore, that in which ds is the square root of a quadratic function of the dx. This case includes space, and is alone considered in what follows. It is called the case of flatness in the smallest parts. Its further discussion depends upon the measure of curvature, the second of Riemann's fundamental conceptions. This conception, derived from the theory of surfaces, is applied as follows. Any one of the shortest lines which issue from a given point (say the origin) is completely determined by the initial ratios of the dx. Two such lines, defined by dx and [delta]x say, determine a pencil, or one-dimensional series, of shortest lines, any one of which is defined by [lambda]dx + µ[delta]x, where the parameter [lambda] : µ may have any value. This pencil generates a two-dimensional series of points, which may be regarded as a surface, and for which we may apply Gauss's formula for the measure of curvature at any point. Thus at every point of our manifold there is a measure of curvature corresponding to every such pencil; but all these can be found when n.[/(n-1)]/2 of them are known. If figures are to be freely movable, it is necessary and sufficient that the measure of curvature should be the same for all points and all directions at each point. Where this is the case, if [alpha] be the measure of curvature, the linear element can be put into the form Entry: A
The most symmetrical treatment of the motion of any point fixed in the top will be found in Klein and Sommerfeld, Theorie des Kreisels, to which the reader is referred for details; four new functions, [alpha], ß, [gamma], [delta], are introduced, defined in terms of Euler's angles, [theta], [psi], [phi], by Entry: 11
if then we can calculate [beta], [alpha], or [beta] - [alpha] for the external shape of the shot, this equation will give the value of [delta] and n required for stability of flight in the air. Entry: W
In 1825, at Union College, _Kappa Alpha_ was organized, copying in style of badge, membership restrictions and the like, its predecessor. In 1827 two other similar societies, _Sigma Phi_ and _Delta Phi_, were founded at the same place. In 1831 _Sigma Phi_ placed a branch at Hamilton College and in 1832 _Alpha Delta Phi_ originated there. In 1833 _Psi Upsilon_, a fourth society, was organized at Union. In 1835 _Alpha Delta Phi_ placed a chapter at Miami University, and in 1839 _Beta Theta Pi_ originated there, and so the system spread. These fraternities, it will be observed, were all undergraduate societies among the male students. In 1910 the total number of men's general fraternities was 32, with 1068 living chapters, and owning property worth many millions of dollars. In 1864 _Theta Xi_, the first professional fraternity restricting its membership to students intending to engage in the same profession, was organized. There were in 1910 about 50 of these organizations with some 400 chapters. In addition there are about 100 local societies or chapters acting as independent units. Some of the older of these, such as _Kappa Kappa Kappa_ at Dartmouth, _IKA_ at Trinity, _Phi Nu Theta_ at Wesleyan and _Delta Psi_ at Vermont, are permanent in character, but the majority of them are purely temporary, designed to maintain an organization until the society becomes a chapter of one of the general fraternities. In 1870 the first women's society or "sorority," the _Kappa Alpha Theta_, was organized at De Pauw University. There were in 1910, 17 general sororities with some 300 active chapters. Entry: FRATERNITIES
(c) The plane moves perpendicularly to the jet. Then [delta] = 90° - [alpha]; cos [delta] = sin [alpha]; and Pu = G/g [omega]u (sin [alpha]/cos [alpha]) (v cos [alpha] - u sin [alpha])². This is a maximum when u = 1/3 v cos [alpha]. Entry: G
But AH = AC cos [delta]/ cos [alpha] = u cos [delta]/ cos [alpha], and therefore HB = v - u cos [delta]/ cos [alpha]. Let [omega] = sectional area of jet; volume impinging on plane per second = Q = [omega](v - u cos [delta]/cos [alpha]) = [omega](v cos [alpha] - u cos [delta])/ cos [alpha]. Inserting this in the formulae above, we get Entry: 157
When the spring is slightly extended by an axial force F, = -R, and there is no couple, so that K vanishes, and [alpha]', r' differ very little from [alpha], r, it follows from these equations that the axial elongation, [delta]x, is connected with the axial length x and the force F by the equation Entry: 65
1. _Arcadian and Cyprian._--As Cyprian was written in a syllabary which could not represent a consonant by itself, did not distinguish between voiced, unvoiced and aspirated consonants, did not represent at all a nasal before another consonant, and did not distinguish between long and short vowels, the interpretation of the symbols is of the nature of a conundrum and the answer is not always certain. Thus the same combination of two symbols would have to stand for [Greek: tote, tode, dote, dothê, tonde, tôde, to, dê]. No inscription of more than a few words in length is found in either dialect earlier than the 5th century B.C. In both dialects the number of important inscriptions is steadily increasing. Both dialects change final [omicron] to [upsilon], [Greek: apo] passing into [Greek: apy]. Arcadian changes the verb ending -[Greek: ai] into -[Greek: oi]. Arcadian uses [delta] or [zeta] for an original _gw_-sound, which appears in Attic Greek as [beta]: [Greek: zellô], Attic [Greek: ballô], "throw." In inflexion both agree in changing -[Greek: ao] of masculine -[alpha] stems into [Greek: au] (Arcadian carries this form also into the feminine -[alpha] stems), and in using locatives in -[Greek: ai] and -[Greek: oi] for the dative, such locatives being governed by the prepositions [Greek: apy] and [Greek: ex] (before a consonant [Greek: es] in Arcadian). Verbs in -[Greek: aô], -[Greek: eô] and -[Greek: oô] are declined not as -[omega], but as -[Greek: mi] verbs. The final [iota] of the ending of the 3rd plural present changes the preceding [tau] to [sigma]: [Greek: pheronsi], cp. Laconian (Doric) [Greek: pheronti], Attic [Greek: pherousi], Lesbian [Greek: pheroisi]. Instead of the Attic [Greek: tis], the interrogative pronoun appears as [Greek: sis], the initial [sigma] in Arcadian being written with a special symbol [koppa]. The pronunciation is not certain. The original sound was _qw_, as in Latin _quis_, whence Attic [Greek: tis] and Thessalian [Greek: kis]. In Arcadian [Greek: kan] the Aeolic particle [Greek: ke] and the Ionic [Greek: an] seem to be combined. Entry: 1
All the errors, except that depending on [alpha], and especially those depending on [gamma] and [delta], can be diminished, without loss of resolving power, by contracting the _vertical_ aperture. A linear error in the spacing, and a general curvature of the lines, are eliminated in the ordinary use of a grating. Entry: FIG
Here let AB be the direction of the beach, A the nearest point to the boat O, and B the point he wishes to reach. Clearly he must land, if at all, between A and B. Suppose he lands at P. Let the angle AOP be [theta], so that OP = a sec[theta], and PB = b - a tan [theta]. If his rate of rowing is V miles an hour his time will be a sec [theta]/V + (b - a tan [theta]) sin [alpha]/V hours. Call this T. Then to the first power of [delta][theta], [delta]T = (a/V) sec²[theta] (sin [theta] - sin [alpha])[delta][theta], so that if AOB > [alpha], [delta]T and [delta][theta] have opposite signs from [theta] = 0 to [theta] = [alpha], and the same signs from [theta] = [alpha] to [theta] = AOB. So that when AOB is > [alpha], T decreases from [theta] = 0 to [theta] = [alpha], and then increases, so that he should land at a point distant a tan [alpha] from A, unless a tan [alpha] > b. When this is the case, [delta]T and [delta][theta] have opposite signs throughout the whole range of [theta], so that T decreases as [theta] increases, and he should row direct to B. In the first case the minimum value of T is also a critical value; in the second case it is not. Entry: A
3. Suppose that the positive value [delta] is an inferior limit to the difference between two real roots of the equation; or rather (since the foregoing expression would imply the existence of real roots) suppose that there are not two real roots such that their difference taken positively is = or < [delta]; then, [gamma] being any value whatever, there is clearly at most one real root between the limits [gamma] and [gamma] + [delta]; and by what precedes there is such real root or there is not such real root, according as [f]([gamma]), [f]([gamma] + [delta]) have opposite signs or have the same sign. And by dividing in this manner the interval ß to [alpha] into intervals each of which is = or < [delta], we should not only ascertain the number of the real roots (if any), but we should also separate the real roots, that is, find for each of them limits [gamma], [gamma] + [delta] between which there lies this one, and only this one, real root. Entry: 3
Suppose we have through Q any other line QT, and let the cosine-inclinations of this to the axes be [alpha]', ß', [gamma]', and [delta] be its cosine-inclination to QP; also let [rho] be the length of the projection of QP upon QT; then projecting on QT we have Entry: 31
2. _Cod. 1 and its Allies_; a group of four MSS. known in Gregory's notation as 1, 118, 131, 209, and in von Soden's as [delta] 50, [epsilon] 346, [delta] 467 and [delta] 457. The dating implied by the latter notation is wrong, as 1 certainly belongs to the 12th, not to the 10th century, and 118 is probably later than 209. It is sometimes quoted as _fam.¹. Fam.¹_ and _fam.¹³_ probably have a common archetype in Mark which is also represented by codd. 28 ([epsilon] 168), 565 ([epsilon] 93, quoted by Tischendorf and others as 2
Quantitative methods are divided into four groups, which we now pass on to consider in the following sequence: ([alpha]) gravimetric, ([beta]) volumetric, ([gamma]) electrolytic, ([delta]) colorimetric. Entry: 5
The physical constants of a given symmetrical top have been denoted in § 1 by M, h, A, C, and l, n, T; to specify a given state of general motion we have G, G´ or CR, D, E, or F, which may be called the dynamical constants; or [kappa], v, w, v1, v2, or f, f´, f1, f2, the analytical constants; or the geometrical constants, such as [alpha], ß, [delta], [delta]´, k of a given articulated hyperboloid. Entry: 25
3. The formal relations between the terms of the series and the differences may be seen by comparing the arrangements (A) and (B) in fig. 1. In (A) the various terms and differences are the same as in § 2, but placed differently. In (B) we take a new series of terms [alpha], [beta], [gamma], [delta], commencing with the same term [alpha], and take the successive sums of pairs of terms, instead of the successive differences, but place them to the left instead of to the right. It will be seen, in the first place, that the successive terms in (A), reading downwards to the right, and the successive terms in (B), reading downwards to the left, consist each of a series of terms whose coefficients follow the binomial law; i.e. the coefficients in b - a, c - 2b + a, d - 3c + 3b - a, ... and in [alpha] + [beta], [alpha] + 2[beta] + [gamma], [alpha] + 3[beta] + 3[gamma] + [delta], ... are respectively the same as in y - x, (y - x)², (y - x)³, ... and in x + y, (x + y)², (x + y)³,.... In the second place, it will be seen that the relations between the various terms in (A) are identical with the relations between the similarly placed terms in (B); e.g. [beta] + [gamma] is the difference of [alpha] + 2[beta] + [gamma] and [alpha] + [beta], just as c - b is the difference of c and b: and d - c is the sum of c - b and d - 2c + b, just as [beta] + 2[gamma] + [delta] is the sum of [beta] + [gamma] and [gamma] + [delta]. Hence if we take [beta], [gamma], [delta], ... of (B) as being the same as b - a, c - 2b + a, d -3c + 3b - a, ... of (A), all corresponding terms in the two diagrams will be the same. Entry: 3
The sound which D represents is the voiced dental corresponding to the unvoiced _t_. The English _d_, however, is not a true dental, but is really pronounced by placing the tongue against the sockets of the teeth, not the teeth themselves. It thus differs from the _d_ of French and German, and in phonetic terminology is called an alveolar. In the languages of India where both true dentals and alveolars are found, the English _d_ is represented by the alveolar symbol (transliterated _d_). Etymologically in genuine English words d represents in most cases _dh_ of the original Indo-European language, but in some cases an original _t_. In many languages _d_ develops an aspirate after it, and this _dh_ becomes then a voiced spirant (ð), the initial sound of _there_ and _that_. This has occurred widely in Semitic, and is found also in languages like modern Greek, where [delta], except after [nu], is always spirant, [Greek: dén] (= _not_) being pronounced like English _then_. As the mouth position for _l_ differs from that for _d_ only by the breath being allowed to escape past one or both sides of the tongue, confusion has arisen in many languages between _d_ and _l_, the best-known being cases like the Latin _lacrima_ as compared with the Greek [Greek: dák-ry]. The English _tear_ and the forms of other languages show that _d_ and not _l_ is the more original sound. Between vowels in the ancient Umbrian _d_ passed into a sound which was transliterated in the Latin alphabet by _rs_; this was probably a sibilant _r_, like the Bohemian _r_. In many languages it is unvoiced at the end of words, thus becoming almost or altogether identical with _t_. As an abbreviation it is used in Latin for the _praenomen_ Decimus, and under the empire for the title _Divus_ of certain deceased emperors. As a Roman numeral (= 500) it is only the half of the old symbol [symbol] (= 1000); this was itself the old form of the Greek [phi], which was useless in Latin as that language had no sound identical with the Greek [phi]. (P. Gi.) Entry: D