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lie between =Ir. By comparing the series (31), (32), (33) with the expansions of cot x, tan x, sec x obtained otherwise, we can calculate the values of U2, U4... V2, V4... and WI, W3.... When U„ has been found, V,, may be obtained from the See also:

formula 2e V" = (2"—I) Un. For See also:

Lord Brounker's series of r, see CIRCLE. It can be got at once Continued by putting a = 1, b = 3, c = 5 .... in See also:

EULER, LEONHARD (1707-1783)

Euler's Factors I I I I See also:

A2(z)

a2 b2. for r. theorem =a—b+c—... =a+b—a+c—b+... See also:

Sylvester gave (Phil.

Meg., 1869) the continued fraction sr I I.2z_33_4 2—I+1+ 1+ 1+ I+...' which is See also:

EQUIVALENT

equivalent to See also:

WALLIS, JOHN (1616-1703)

Wallis's formula for sr. This fraction was originally given by Euler (See also:

COMM

Comm. Acad. Petrojol. vol. xi.) ; it is also given by Stern (in Crelle's Journ. vol. x.). 3o. It may be shown by means of a transformation of the series See also:

sin x x x2 x2 x3 Continued for See also:

cos x and x that tan x=-- 3 — 5 — 7 — Fractions x= implies the antecedent existence of much simpler types, and refers I the origin of See also:

LIFE

life to a date immeasurably distant from that at which we have actual See also:

PROOF (in M. Eng. preove, proeve, preve, &°c., from O. Fr . prueve, proeve, &c., mod. preuve, Late. Lat. proba, probate, to prove, to test the goodness of anything, probus, good)

proof of the existence of See also:

animal and See also:

VEGETABLE

vegetable organisms. However different in structure Trilobites may be, they all agree in possessing a See also:

head-See also:

shield usually semi-circular in shape, which results from the See also:

FUSION

fusion of apparently five segments, and bears, except in some See also:

BLIND, MATHILDE (1841—1896)

blind forms, a pair of large reniform See also:

compound eyes like those of the See also:

king-crab (Xiphosura). This head-shield is succeeded by a varying number of See also:

FREE

free segments, each of which consists of a See also:

MEDIUM

medium See also:

CONVEX

convex tergal piece and a pair of arched lateral plates, the pleura, of which there is one on each See also:

side. The terga and pleura of each individual segment overlap those of the segment that serially succeeds it. The See also:

mid-region of the See also:

BODY

body, composed of jointed segments, is followed by a larger or smaller region consisting of fused segments and termed the pygidium or caudal shield, which in some cases is as large as the head-shield itself, in other cases much smaller. When the pygidium is large and composed of many segments, the number of free body segments is correspondingly reduced, and See also:

VICE

vice versa. It is with respect to this number of segments that respectively constitute the pygidium and the mid-region of the body that Trilobites differ most markedly from each other; and it is a singular fact that the extremes in structural organization in this particular to be met with in the Trilobita are found side by side in strata of Cambrian See also:

AGE (Fr. age, through late Lat. aetaticum, from aetas)

age.

In Paradoxides, for example, there are about twenty freely movable segments followed by a very See also:

SHORT, FRANCIS JOB (1857– )

short and small pygidium, whereas in Agnostus the freely movable segments are reduced to two and the pygidium is as large as the cephalic shield. In this genus the number of segments composing the pygidium is obscured, as also it is in the genus Illaenus, which has as many as ten movable segments pre-ceding the large semi-circular pygidium; but in such forms as Ogygia and Asaphus, which have about eight free segments, the sutural lines on the pygidium indicate that it is composed of about a dozen or more segments. Somewhat resembling Agnostus is Microdiscus, with four movable segments and a large pygidium consisting of about five fused segments, the lines of See also:

union between the latter being clearly indicated. The tergal and pleural elements of the pygidium are generally well marked. They are also well marked on the cephalic shield, the tergal elements being represented by a median axial elevated See also:

AREA

area showing indistinct signs of segmentation, and a lateral unsegmented See also:

PLATE

plate, the gena, which carries the eyes. The posterolateral angles of the gena are commonly produced into spiniform processes, which may project backwards beyond the See also:

MIDDLE

middle of the body as in Paradoxides, or considerably beyond its posterior termination as in Trinucleus or Ampyx. The latter is further remarkable for having the median area of the head-shield, the flabellum, produced into an anteriorly directed spike. For many years only the dorsal See also:

SURFACE

surface of Trilobites was known, nothing having been ascertained of the ventral surface and appendages. Comparatively recently, however, specimens have been obtained with the ventral surface exposed, revealing the number and structure of the limbs. A pair of the latter was articulated to the sides of a moderately wide dorsal plate on each segment of the body, and similar limbs were attached to the ventral surface of the head-shield behind the mouth. Each of these limbs was two-branched, the See also:

EXTERNAL

external See also:

BRANCH (from the Fr. branche, late Lat. branca, an animal's paw)

branch consisting of a slender fringed flagellum possibly See also:

RESPIRATORY

respiratory in See also:

FUNCTION

function, and the inner of a normal jointed See also:

AMBULATORY (Med. Lat. ainbulatorium, a place for walking, from ambulare, to walk)

ambulatory See also:

leg. These two branches arose from a See also:

COMMON

common basal segment or coxa, the inner surface of which was produced into a strong See also:

PROCESS

process underlying the external area.

In the region of the mouth the basal segments were armed with See also:

TEETH (O.E. Eel); plural of tooth, O.E. top)

teeth and subserved the purpose of mastication. As in all Arachnida there is only a single pair of appendages in front of the mouth, and these were one-branched, See also:

long and filiform and acted as antennae. Under the pygidium or caudal shield the appendages were much shortened, and their See also:

main branch consisted of broader and flatter segments than those of the preceding limbs. Such was the structure of the appendages in Trilobites belonging to the genus Triarthrus; but considering the See also:

GREAT

great structural See also:

DIFFERENCES, CALCULUS OF (Theory of Finite Differences)

differences that obtain between Triarthrus and many other genera, it would be rash to assume that there were not corresponding differences in the structure of the limbs. It must not indeed be assumed that those of the first pair were in all cases antenniform. It is probable that no .satisfactory See also:

CLASSIFICATION (Lat. classis, a class, probably from the root cal-, cla-, as in Gr. icaMce, clamor)

classification of the Trilobites will be proposed until the limbs of most of the genera have been examined. Up to the See also:

PRESENT

present time all attempts to arrange the genera in natural and definable See also:

groups have failed to meet with See also:

general approval; and this See also:

criticism must be extended to See also:

Beecher's subdivision of the class into three orders, named Hypoparia, Proparia and Opisthoparia, based upon the See also:

FORM (Lat. forma)

form and position of a groove, the so-called genal suture, which marks the lateral portion of the head-shield. In the See also:

MAJORITY (Fr. majorite; Med. Lat. majoritas; Lat. major, greater)

majority of Trilobites this groove passes backwards from the anterior or anterolateral edge of this plate to its posterior or postero-lateral border, dividing it into an inner portion continuous with the flabellum and fused tergal regions, and an See also:

OUTER

outer portion bearing the See also:

eye. Those genera, like Paradoxides, Olenus, Asaphus, Phillipsia and others, in which this groove cuts the posterior edge of the head-shield on the inner side of its See also:

ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form, angus, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

angle are referred to the Opisthoparia; those, like Dalmanites and Phacops, in which it cuts the lateral border in front of the posterior angle, belong to the Proparia. But in certain genera, like Conocoryphe, Calymmene and Triarthrus, it cuts the margin of the head-shield so close to the posterior angle that the distinction between the two groups practically breaks down. To the Hypoparia belongs a comparatively small number of genera, like Trinucleus and Aquastus, in which this groove or genal suture is beneath the margin of the head-shield and does not appear upon its upper surface. In external form Trilobites are not unlike Isopod Crustaceans, especially the terrestrial See also:

SPECIES

species commonly called " woodlice "; and until the nature of their appendages was known, it was thought by some authorities that the two groups might be related.

Like the woodlice they were capable of See also:

ROLLING

rolling themselves up into a See also:

ball, many specimens having been found fossilized in this See also:

state, with the pygidium pressed tightly against the head-shield. There is very little doubt that they lived at the bottom of the See also:

sea, feeding upon See also:

WORMS

worms or other soft marine organisms, crawling slowly about the sandy or muddy bottom and burying themselves beneath its surface when danger threatened. That these animals were widely distributed in former times is proved by their occurrence at the present See also:

day in palaeozoic fossiliferous strata both of the See also:

NORTHERN

northern hemisphere and of See also:

AUSTRALIA

Australia; and despite the fact that their remains have not been found in rocks of the Mesozoic or Kainozoic epochs, it was conceived to be possible that living specimens might be dredged from the sea-See also:

FLOOR (from O. Eng. flor, a word common to many Teutonic languages, cf. Dutch vloer, and Ger. Flur, a field, in the feminine, and a floor, masculine)

floor during the exploration of the ocean depths undertaken by the " Challenger " expedition. Needless to say this faint See also:

hope was not See also:

BORNE, KARL LUDWIG (1786–1837)

borne out by results. (R. I.

End of Article: TRILOBITES

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