notation 
something further Is to be added or subtracted. The sign 
, called plug or minus, is ordinarily used in a disjnne 
tive sense in writing the root of a quadratic equation. 
Thus, if x" + x = 1, wo write x = i (1 !_ y R), meaning 
that thi! equation is satisfied only by the two values X = 
i (1 + V 6 ) "d x = J (1 |/ 5). The sign is also 
used in ustr ny, geodesy, etc., after a value determined 
by observation, t<> introduce the probable error of that de- 
termination. Summation is also signified by the letter i. 
Thus, 2i(l/i) means that in the expression 1/i all the 
whole numbers from 1 to n inclusive are to be successive- 
ly substituted for i mid the resulting values added to- 
gether to give the quantity denoted by the expression. 
When the limits are not indicated, the lower one is to be 
understood as constant, and generally zero, and the upper 
one as one less than the actual value of the variable. For 
example, U we write (2* + 1) = x'i, this signifies 
* S,. (2 I 1) = *3. 
o 
In like manner. A is used to signify the difference, or the 
amount by which the quantity written after it would be in- 
creased by increasing the variable by unity. The variable 
may be Indicated by a subjacent letter ; thus, *x* = 
(x + iy x-* ; but A^oP 1 = x' + i x* = (x IJz-*. The 
Sroduct of two quantities is denoted by writing them in 
leir order, cither directly, or with an Interposed cross (x) 
or dot (.) ; thus, :i x o = 3 . o = So. A Quotient is usually 
denoted by one of the signs T or :or/, with the dividend be- 
fore it anu the divisor after it, or by a horizontal line with 
the dividend above and the divisor below. A continued 
product is also written with II, just as a summation is writ- 
ten with i: but when the limits are not indicated, the lower 
one is constant, and generally unity, and the upper one the 
actual value of the variable. A positive whole number with 
the mark of admiration (!) after it denotes the continued 
product of all numbers from 1 up to that number inclusive ; 
thus, 4! = 24. Instead of the mark of admiration, a right- 
angled line beneath and at the left of the number is some- 
times used : as, \*~ A power of a quantity is denoted by 
writing the exponent to the right and above the base; thus, 
z3 = x . x . x. This notation is extended to symbols of op- 
eration. Thus, A-M = AAu; and A lu = 2, because u 
AA 1 = ASM. If the exponent is Included in parenthe- 
ses, the quantity denoted Is the continued product of a 
number of factors equal to the exponent, one factor being 
the base, and the others the results of successive subtrac- 
tions of 1 from the base ; thus, ( ;1 ) = x (x \)(x 2). A 
root is denoted either by a fractional exponent, or by the 
sign V written before the base, with the index above and 
to the left ; thus, y 8 = 2. If the index Is omitted, it is 
understood to be 2. One of the most important parts of 
algebraical notation is the use of parentheses, ( ), square 
brackets, [ 1, braces, j [, and vincula or horizontal lines 
above the expressions, to signify that the symbols so In- 
cluded are to be treated as signifying one quantity. Thus, 
(8 + 2) x 5 = 25, but 8 + (2x5) = 13. Functions are 
usually denoted by operative symbols, especially /, F, 
$, *, written before the variable, the latter being often 
inclosed in parentheses. If there are several variables, 
these are inclosed in one parenthesis and separated by 
commas, as F (x, y). Various special functions have spe- 
cial abbreviations, as log for logarithm, sin for sine, cos 
for cosine, tan for tangent, cot for cotangent, sec for se- 
cant, cosec for cosecant, vsin for versed sine, sinh for hy- 
perbolic sine, am for amplitude, sn for sine of the ampli- 
tude, en for cosine of the amplitude, etc. (For the special 
notation of matrices, determinants, graphs, and groups, see 
those words.) A differential is expressed by d before the 
function, and a partial differential is now generally writ- 
ten with <' instead of d; the variable is indicated, if ne- 
cessary, by a subjacent letter. A variation is expressed by 
a S before the varying quantity. A differential coefficient 
is most frequently expressed fractionally as a ratio of dif- 
ferentials, or by -;-, etc. , written before the function. But 
the capital D is often used : thus, D'z^ = yx* 1, and 
D>*> = log x . X-?. Differentiation relatively to the time 
is frequently expressed by accents : thus, = D/s and " = 
Dra 1 . Dots over the letters are also used instead of the 
accents, this being the original Huxional notation of New- 
ton. The differential coefficients of a function are fre- 
quently denoted by accents attached to the operational 
symbols : thus, /"* = DJ/z. A number of other differ- 
ential operations are indicated by special operational sym- 
bols, as r for Laplace's operator. The integral of an ex- 
pression Is written with the sign /, introduced by Leib- 
nitz, before the differential. The limits of a definite in- 
teitnil :ire written above and below this sign. Besides 
these notations, there are many others peculiar to differ- 
ent branches of mathematics. 
3f. Etymological signification ; etymology. 
The notation of a word is when the original thereof is 
sought out, and consisteth In two things: the kind and 
the figure. B. Jonson, Eng. Grammar, viii. 
Conscience Is a Latin word, and, according to the very 
notation of It, imports a double or joint knowledge. South. 
4. In music, the act, process, or result of indi- 
cating musical facts by written or printed char- 
acters. As a process and a science, musical notation is 
a branch of semiotics or semiography in general. ..Vo- 
tatwn is also used as a collective term for all the signs for 
musical facts taken together. Notation, whether regarded 
as a science or as a body of visible characters, may be di- 
vided into notation of pitch, of duration, of force, of style, 
etc. The various historic systems of notation are more par- 
ticular about I'itch than about the other matters, (a) The 
absolute and relative pitch of tones has been represented 
by letters, by neumes, by syllables, by numerals, by a staff, 
and by more than one of these methods at (Mice. The an- 
cient lireeks and Romans used their alphabets, assigning 
sometimes a separate tetter or similar character to each 
tone of their tonal systems, and sometimes usingonly seven 
letters, which were repeated for successive octaves. The 
medieval notations included all the different methods, 
used both separately and in conjunction, letter-names be- 
ing derived from the ancient notations, neumes appearing 
4023 
early from an unknown source, and solmlzatfon and the 
M;itt--'\sirni tiring invented ;md developed from alM>ut the 
eichthur ninth century. Modern nutations include all va- 
rieties except neumes. See letter-name, neutne, mlmization, 
numeral, keyboard, scale, tta/, etc. (b) The absolute and 
relative duration of tones has been ranch leu fully indi- 
cated than pitch. The ancient and medieval systems were 
decidedly defective in this regard. The appearance about 
the twelfth century of mensurable music necessitated the 
use of characters having a definite metrical value ; hence 
came the note-system, which was combined with the staff, 
:ni'l ulso the various systems of tablature. In modern mu- 
sic two methods are used notes whose shape Indicates 
relative time- value, ami a kind of tablature peculiar to the 
tunic sol-fa system. (See note, tablature, tunic sol-fa (un- 
der tonic), etc. ) Furthermore, the general tempo of a piece 
or passage is indicated by such Italian terms as graoc.ada- 
tjio, umlnntr, moderate, allegro, vivace, presttt, etc. Altera- 
tions of tempo during a piece are indicated by accelerando, 
piu MOJWO, xtringendo, rilardando,ritenuto, calando, etc. The 
metrical treatment of individual tones is marked by ttae- 
cato, legato, etc. () The absolute and relative force or 
accent of tones is still less fully indicated than pitch or 
duration. Vertical lines called tart have been used since 
medieval times to indicate rhythmical and metrical sec- 
tions or measures, each of which begins with a primary 
accent. In modern music various words and arbitrary 
signs are used, as forte, piano, crescendo (<" X diminuendo 
( ^, marcando. (a) Other signs of various practical 
import are the brace, repeat, da capo, dal scyno, double bar, 
slur, etc. See these words, (e) The general style of a piece 
or passage is often indicated In modern music by such 
terms as ad libitum, agitato, arpeggio, cantabile, tspressioo, 
sostenuto, con brio. (/) Specific directions about per- 
formance by the voice or an Instrument also occur, as 
mezza voce, arcato, portamento, divisi, mono sinistra, pizzi- 
cato, -ni (oUava), pedal, and many others. All these ver- 
bal marks are translated Into different languages, and are 
subject to modification for particular effects, (g) Mod- 
ern music, following the later medieval music, also em- 
ploys to some extent a kind of numerical shorthand for 
harmonic facts. See thorough-bass, and figured bass (under 
(wu3)t_ Alphabetic notation, in musu. See def. 4 (a). 
Architectural notation, a method adopted of placing 
signs to figures when marking dimensions on drawings: as 
' for feet, " for inches, and '" for parts, etc. Chemical no- 
tation, a system of abbreviating and condensing state- 
ments of the chemical composition of bodies, and of their 
changes and transformations, by means of symbols. See 
chemical formula, under chemical. Decimal notation. 
3ee decimal NeumatlC notation, in mutic. See def. 4 
(a), above, and also neume. Numerical notation, in mu- 
sic. See def. 4 (g\ above, and numerical. Staff-nota- 
tion, In music. See def. 4 (a) and (b\ above, and also staff 
and note. Tonic sol-fa notation, in music. See tonic. 
notator (no-ta'tor), n. [< ML. notator, < L. no- 
tare, note: see note 1 ,.] An annotator. [Rare.] 
The notator Dr. Potter In his epistle before It to the 
reader saith thus, Totum opus, &e. Wood, Athene Oxon. 
notch (noch), n. [An assibilated form of nock.] 
1. A nick or indentation; a small hollow or 
nick cut or sunk in anything, as in the end of 
an arrow for the reception of the bowstring. 
From his rug the skew'r he takes, 
And on the stick ten equal notches makes. 
Suift, Miscellanies. 
The Indented stick that loses day by day 
Notch after notch, till all are smootn'd away. 
Cmcper, Tirocinium, 1. 560. 
2. In carp., a hollow cut in the face of a piece 
of timber for the reception of another piece. 
3. A narrow defile or passage between moun- 
tains ; or, more properly, the entrance to such a 
defile, when it is nearly closed by precipices or 
walls of rock on either hand. The word Is appar- 
ently limited in use to the region of the White Mountains 
in New Hampshire and of the Adirondack*, and has nearly 
the same meaning as gap in the central parts of the Ap- 
palachian range. [U. S.j 
They landed, and struck through the wilderness to a gap 
or notch of the mountains. Irring. 
4. A step or degree; a grade. [Colloq.] 5. 
A point in the game of cricket. [Rare.] 
A match at cricket between the gentlemen of Hampshire 
and Kent on the one side and All England on the other 
[1788]. She former won, says the "Annual Register," by 
" twenty%ur notches." Fortnightly Rev., N. 8., XUII. 377. 
6. In zool. and mint., an incision or incisnre; 
an emargination : as, the interclavicular notch, 
the depression over the breast-bone between 
the prominent ends of the clavicles. 7. In 
armor, the bouche of a shield. Anterior notch of 
the liver, a deep angular Incisure In the front border of 
the liver, between the right and left lobes. Also called 
umbilical or interlnbular incisure or notch. Clavicular 
notch, one of the superior lateral depressed surfaces of 
the presternmn, for articulation with the clavicles. Co- 
tylold, cranlofaclal, dicrotic notch. See the adjec- 
tives. Ethmoidal notch,the mesial excavation between 
the orbital plates of the frontal bone, for the reception of 
the ethmoid bone. Great scapular notch, the notch 
formed by the neck of the scapula and the acromion pro- 
cess. Intercondylar notch, the notch or fossa between 
the femoral condyles behind. Interlobular notch. 
See anterior notch of the liver. Intervertebral notch, 
a concavity on the upper and lower borders of the pedicle, 
forming, when in apposition with those of the contiguous 
vertebra 1 , the intervertebral foramina, Jugular notch, 
ft notch in front of the jugular process of the occipital bone, 
which contributes, with one on the temporal bone, to form 
the jugular foramen. Lacrvmal notch, an excavation 
on the internal border of the orbital surface of the max- 
illa, for the reception of the lacrymal bone. Nasal 
notch, (a) A serrated surface of the frontal bone, for ar- 
note 
ticnlatlon of the nasal and superior maxillary bone*. (6) 
The large notch of the maxilla that forms the lateral and 
lower boundary of the entrance to the nasal cavity. 
Notch Of Rlvlnl, a small notch in the upper anterior part 
nf Hit: bony ring to which tin: t)inpanlc membrane U at- 
tached. Al>' mic notch. Notch of the con- 
cha, the incisura inU'rtragica, or notch between the tra- 
gusand theaiititragUB. Notch Of the kidney, tliehilmn 
or porta renis. Popliteal notch, a shallow depression 
bet cen the tlliial tnberoslties behind.- Posterior notch 
of the liver, a wide concave recess between the right and 
left lobes of the 11 ver.embraclng the cruraof the diaphragm, 
the cava, the aorta, and the esophagus. Ptery gold 
notch, the angular cleft between the two plates of the 
pterygold process, closed by the palate-bone. Also called 
incimra pUrygoidea. SdiMc notch, one of two notches 
on the posterior border of the hip bone, the great (or illo- 
sclatlc) and the small The great sciatic notch Is between 
the posterior inferior spine of the Ilium and the spine 
of the ischium, and U converted into the great sacro 
sciatic foramen by the sacrosciatic ligaments ; the small 
sciatic notch Is between the spine and the tnberoslty of 
the iachfum, and is converted into a foramen by the same 
ligaments. Slgmoid notch, the excavation between the 
condyleand the coronold process of the mandible. Sphe- 
nopalatlne notch, a notch between the sphenoids! and 
orbital processes of the palate-bone, converted into the fora- 
men of the same name by the sphenoid bone. Supra- 
orbital notch, a notch at the inner part of the orbital 
arch, transmitting the supraorbital nerve and artery. It 
is often a foramen. Suprascapular notch, the notch on 
the superior border of the scapula, at the base of the 
coracoid process, converted into a foramen by a ligament or 
a spiculum of bone. Suprasternal notch, the notch or 
depression at the upper end of the sternum, between the 
sternal ends of the sternoclidomastold muscles. The 
top notch, the highest grade or degree of anything : as, 
Otttop notch of fashion or elegance. [Colloq. j Tympanic 
notch. Same as notch of Kieini. Umbilical notch. See 
anterior notch of the lioer. 
notch (noch), v.t. [(notch, n. Of. nock, v.~) 1. 
To cut a notch or notches in; indent; nick; 
hack: as, to notch a stick. 
Before Corloli he scotched him and notched him like a 
carbonado. Shak., Cor., iv. S. 199. 
2. To place in a notch ; fit to a string by the 
notch, as an arrow. 
Mark how the ready hands of Death prepare ; 
His bow is bent, and he hath notch'd his dart. 
Quartet, Emblems, L 7. 
3. In cricket, to mark or score ; have as score 
the number of. [Slang.] 
In short, when Dumkins was caught out, and Fodder 
stumped out, All-Muggleton had notched some fifty-four, 
while the score of the Dingley Dellers was as blank as 
their faces. Dickens, Pickwick, vil. 
notch-block (noch'blok), n. Same as snatch- 
block. 
notch-board (noch'bord), n. In carp., same as 
bridge-board. 
notch-eared (noch'erd), a. Having emargi- 
nate ears: as, the notch-cared bat, Vespertilio 
emarginatus. 
notched (nocht), a. 1. Having a notch or 
notches; nicked; indented. 
The middle claw of the heron and cormorant Is toothed 
and notched like a saw. Paley, Nat. Theol., xlll. 
2. Closely cut; cropped, as hair: applied by 
the Cavaliers to the Roundheads. 
She had no resemblance to the rest of the notch'd ras- 
cals. Sir R. Howard, The Committee, i. (Danes.) 
3. In zool., having one or more angular inci- 
sions in the margin; emarginate. 4. In hot., 
very coarsely dentate, the upper side of the 
teeth being nearly horizontal, as in the leaves of 
Shun toiicodendron. Notched falcon. See falcon. 
notchel (noch'el), r. t. See nocltel. 
notching (noch'ing), n. [Verbal n. of notch, 
p.] 1. A notch or series of notches. 2. In 
cngin., same as gulleting. 3. In carp., a simple 
method of joining timbers in a frame, either 
by dovetails or by square joints or lap-joints. 
Calking, halving, and scarfing are forms of it. 
notching-adz (uoch'ing-adz), n. A light adz 
with a bit either of large curvature or nearly 
straight, used for notching timbers in making 
gains, etc. E. H. Knight. 
notching-machinefnocb/ing-ma-shen'), n. 1. 
In sheet-metal working, a form' of stamping- 
press for cutting the corner notches in making 
boxes, hinges, and other shapes of sheet-metal. 
notchweed (noch 'wed), n. An ill-smelling 
herb, Cheno/iodiiim Tulraria, of the northern 
parts of the Old World. Also called stinking 
</<><ixcfoot and doffs-orach. 
notchwing (uoch'wing), n. A European tpr- 
trieid moth, Bliacodia caudana : an English 
collectors' name. 
note 1 (not), H.I [Early mod. E. also noat; < ME. 
note, noote, a note. mark, point (not from the 
rare AS. not, a mark, note),< OF. note, F. note = 
Sp. Pg. It. nota, < L. worn, a mark, sign, criti- 
cal mark or remark, note, < noscere, pp. notvs, 
know: see know 1 . Hence note 1 , v., notary 1 , 
etc. Cf. note 1 , a.] 1. A mark or token by 
