j{m\r\-\-m\r\->r...-Vmlr^;), ... (2)
where r^, r^, . , . r, are the probable errors of the several
observed quantities.
In particular, if the n quantities have the same probable error
r, the probable error of their sum is rV«- The probable error
of their arithmetical mean, which is — of this sum, is therefore
' n
4—. This result agrees with that found in Art. 64, where,
t^n "
* The fact that the law of facility thus reproduces itself has often been
regarded as confirmatory of its truth. This property of the law «~
results from its being a limiting form for the facility of error in the linear
function Z, when n is large, whatever be the forms of the facility functions
for Xi, Xi, . . . X,. Compare the foot-note on page 49, and see the
memoir there referred to. It follows that " we shall obtain the same
law g—^'"' (for a single observed quantity) if we regard each actual
eiror as formed by the linear combination of a large number of errors
due to different independent sources."
§ VI.] NON-LJM£AR FUNCTIONS. 73
however, the n quantities were all observed values of the same
quantity, and the arithmetical mean was under consideration by
virtue of its being the most probable value in accordance with
the law of facility.
90. It is to be noticed that in formula (2) it is essential that
the probable errors ^i , rj, . . . r„ should be the results of inde-
pendent determinations. For example, in the illustration given
in Art. 88, we have ki^^ f + p, whence we should expect to find
(prob. err. of hif = (prob. err. of (pf + (prob. err. of ^)'' ;
but it will be found that this is not true when the probable
errors of ^l.
13. What is the probable error of the area of the rectangle
whose sides measured as in the preceding example are ^'land s^ ?
14. A line of levels is run in the following manner : the back
and fore sights are taken at distances of about 200 feet, so that
there are thirteen stations per mile, and at each sight the rod is
read three times. If the probable error of a single reading is
0.01 of a foot, what is the probable error of the difference of level
of two points which are ten miles apart ? .093.
15. Show that the probable error of the weighted mean of
observed quantities has its least possible value when the weights
are inversely proportional to the squares of the probable errors
of the quantities, and that this value is the same as that given in
Art. 68 for the case of observed value of tiie same quantity.
VII.
The Combination of Independent Determinations of
THE Same Quantity.
The Distinction between Precision and Accuracy.
ijl2. We have seen in Arts. 63 and 67 that the final determi-
nation of the observed quantity derived from a set of observations
follows the exponential law of the facility of accidental errors.
The discrepancies of the observations have given us the means
of determining a measure of the risk of error in the single
observations, and we have found that the like measure for the
final determination varies inversely as the square root of its
weight compared with that of the single observation. Since
this weight increases directly with the number of constituent
observations, it is thus possible to diminish the risk of error
indefinitely ; in other words, to increase without limit the pre-
cision of our final result.
93. It is important to notice, however, that this is by no means
the same thing as to say that it is possible by multiplying the
number of observations to increase without limit the accuracy
of the result. Ihe precision of a determination has to do only
with the accidental errors ; so that the diminution of the prob-
able error, while it indicates the reduction of the risk of such
errors, gives no indication of the systematic* errors (see Art. 3)
*The term systematic is sometimes applied to errors produced by a
cause operating in a systematic manner upon the several observations,
thus producing discrepancies obviously not following the law of accidental
errors. Usually a discussion of these errors leads to the discovery of
their cause, and ultimately to the corrections by means of which they may
be removed. All the remaining errors, whose causes are unknown, are
generally spoken of as accidental errors ; but in this book the term acci-
dental is applied only to those errors which are variable in the system of
observations under consideration, as distinguished from those which have
a common value for the entire system.
§VII.] PRECISION AND ACCURACY. JJ
which are produced by unknown causes affecting all the obser-
vations of the system to exactly the same extent.
The value to which we approach indefinitely as the precision
of the determination is increased has hitherto been spoken of
as the " true value," but it is more properly the precise value
corresponding to the instrument or method of observation
employed. Since the systematic error is common to the whole
system of observations, it is evident that it will enter into the
final result unchanged, no matter what may be the number of
observations ; whereas the object of increasing this number is
to allow the accidental errors to destroy one another. Thus the
systematic error is the difference between the precise value,
from which accidental errors are supposed to be entirely elimi-
nated, and the accurate or true value of the quantity sought.
94. Hence, when in Art. 64 the arithmetical mean of n obser-
vations was compared to an observation made with a more
precise instrument, it is important to notice that this new
instrument must be imagined to lead to the same ultimate
precise value, that is, it must have the same systematic error as
the actual instrument, whereas in practice a new instrument
might have a very different systematic error.
Again, in the illustration employed in Art. 64, where the final
determination of an angle is given as 36° 42'.3 ± i'.22, the
" true value," which is just as likely as not to lie between the
limits thus assigned, is only the true value so far as the instru-
ment and method employed can give it ; that is, the precise value
to which the determination would approach if its weight were
increased indefinitely.
95. A failure to appreciate the distinction drawn in the
preceding articles may lead to a false estimate of the value
of the method of Least Squares. M. Faye in his "Cours
d'Astronomie " gives the following example of the objections
which have been urged against the method: "From the
discussion of the transits of Venus observed in 1761 and 1769,
M. Encke deduced for the parallax of the sun the value
8".57ii6±o".o37o.
78 INDEPENDENT DETERMINATIONS. lArt. 95
In accordance with this small probable error it would be a
wager of one to one that the true parallax is comprised between
8". 53 and 8". 61. Now we know to-day that the true parallax
8". 813 falls far outside of these limits. The error, o".24i84, is
equal to 6.536 times the probable error o".037. We find for
the probability of such an error o.ooooi. Hence, adhering to
the probable error assigned by M. Encke to his result, one could
wager a hundred thousand to one that it is not in error by
0.24184, and nevertheless such is the correction which we are
obliged to make it undergo."
Of course, as M. Faye remarks, astronomers can now point
out many of the errors for which proper corrections were not
made ; but the important thing to notice is that, even in Encke's
time, the wagers cited above were not authorized by the theory.
The value of the parallax assigned by Encke was the most
probable with the evidence then known, and it was an even wager
that the complete elimination of errors of the kind that produced
the discrepancies or contradictions among the observations could
not carry the result beyond the limit assigned ; but the existence
of other unknown causes of error and the probable amount of
inaccuracy resulting from them is quite a different question.
Relative Accidental and Systematic Errors.
96. Let us now suppose that two determinations of a quantity
have been made with the same instrument and by the same
method, so that they have the same systematic error, if any ; in
other words, they correspond to the same precise value. The
difference between the, two results is the algebraic difference
between the accidental errors remaining in the two determi-
nations; this may be called their relative accidental error.
Regarding the two determinations as independent measure-
ments of two quantities, if r, and ra are their probable errors,
that of their difference is V iA + ^0 ; and, since this difference
should be zero, the relative error is an error in a system for
which the probable error is
§VII.] ACCIDENTAL AND SYSTEMATIC ERRORS. 79
For example, if the determination of an angle mentioned in Art.
94 is the mean often observations, it is an even wager that the
mean of ten more observations of the same kind shall differ
from 36° 42'.3 by an amount not exceeding i'.22 X V 2 or i'.73
Again, r being the probable error of a single observation, the
probable error of the mean of n observations is — t— , but the
discrepancy from this mean of a new single observation is as
likely as not to exceed
\/(^ +''')• ^^^*'^' V
« + I
97. If, on the other hand, the two determinations have been
made with different instruments or by a different method,
they may involve different systematic errors ; so that, if each
determination were made perfectly precise, they would still
differ by an amount equal to the algebraic difference of their
systematic errors. Let this difference, which may be called the
relative systematic error, be denoted by 8. Then, d denoting
the actual difference of the two determinations, while 8 is the
difference between the corresponding precise values, we may
put
^ = 3 + X,
in which x is the relative accidental error.
The Relative Weights of Independent Determinations.
98. In combining values to obtain a final mean value, we have
hitherto supposed their relative weights to be known or assumed
beforehand, as in Arts. 75 and 77. Since the squares of the
probable errors are inversely proportional to the weights, (Arts.
66 and 68,) the ratios of the probable errors both of the con-
stituents and of the mean are thus known in advance, and it
*This does not apply to the residuals of the original « observations,
because in taking a residual the mean is not independent of the single
observation with which it is compared.
80 INDEPENDENT DE TERMINA TIONS, [An. 98
only remains to determine a single absolute value of a probable
error to fix them all. In this process it is assumed that the
values have all the same systematic error.
But, when the determinations are independently made, their
relative weights are not known, and their probable errors have
to be found independently. If now it can be assumed that the
systematic errors are the same, so that there is no relative
systematic error, the weights may be taken in the inverse ratio
of the squares of the probable errors.
99. To determine whether the above assumption can fairly be
made in the case of two independent determinations whose
probable errors are r^ and r, , it is necessary to compare the
difference d with the relative probable error i^(rl + rf), Art. 96.
If d is small enough to be regarded as a relative accidental
error, it is safe to make the assumption and combine the deter-
minations in the manner mentioned above.
As an example, let us suppose that a certain angle has been
determined by a theodolite as
24° 13' 36"±3"-i,
and that a second determination made with a surveyor s transit
24° I3'24"±i3".8.
In this case r, = 3.1, r, = 13.8 and d= 12. It is obvious that
a relative accidental error as great as d may reasonably be
expected. (In fact the relative probable error is 14. i ; and, by
Table II, the chance that the accidental error should be at least
as great as 12 is about .57.) We "may therefore assume that
there is no relative systematic error, and combine the determi-
nations with weights having the inverse ratio of the squares of
the probable errors. This ratio will be found, in the present
case, to be about 20 : i, and the corresponding weighted mean
found by adding ^ of the difference to the first value, is
24° 13' 35"43-
100. It appears doubtful at first that the valae given by the
§V1I,] CONCORDANT DETERMINATIONS. 8l
theodolite can be improved by combining with it the value
given by the inferior instrument. The propriety of the above
process becomes more apparent, however, if we imagine the
first determination to be the mean of twenty observations made
with the theodolite ; a single one of these observations will then
have the same weight and the same probable error as the second
determination. Now the discrepancy of this new determination
from the mean is such as we may expect to find in a new single
observation with the theodolite. We are therefore justified in
treating it as such an observation, and taking the mean of the
twenty-one supposed observations for our final result.
lOI* The probable error of the result found in Art. 99 of
course corresponds with its weight; thus, denoting it by R, we
have ^=i^f\, whence R = 3" .03, and the final result is
24° 13' 35".43 ± 3"-03.
In general, ri and ra being the given probable errors, that oi
the mean is given by
rl + r\'
Determinations which, considering their probable errors, are
in sufficient agreement to be treated as in the foregoing articles
may be called concordant determinations. They correspond to
the same precise value of the observed quantity, and the result
of their combination is to be regarded as a better determination
of the same precise value.
The Combination of Discordant Determinations.
102. As a second illustration of determinations independently
made, let us suppose that a determination of the zenith distance
of a star made at one culmination is
14° 53' i2".i ±o".3,
and that at another culmination we find for the same quantity
14° 53' i4"-3 ± o".5.
In this case we have d = 2.2. This is about 3.8 times the rela
tive probable error whose value is o".58.
82 IMDEPENDENT DETERMINATIONS. [Art. 102
From Table II we find that the probability that the relative
accidental error bhould be as great as d is only about i in 100.
We are therefore justified in assuming that the difference rf is
mainly due to errors peculiar to the culminations. In other
words, we assume that, could we have obtained the precise
values corresponding to the two culminations, (by indefinitely
increasing the number of observations at each,) they would still
be found to differ by about 2". 2. Supposing now that there is
no reason for preferring one of these precise values to the other,
we ought to take their simple arithmetical mean for the final
result ; and, since the two given values are comparatively close
to the precise values in question, we may take their arithmetical
mean, which is
14° 53' i3"-2,
for the final determination.
103. Determinations like those considered above, whose
difference is so great as to indicate an actual difference between
the precise values to which they tend, may be called discordant
determinations. The discordance of the two determinations
discloses the existence of systematic errors which were not
indicated by the discrepancies of the observations upon which
the given probable errors were based. In combining the deter-
minations, these systematic errors ar^ treated as accidental
errors incident to the two determinations considered as two
observed values of the required quantity. In fact, it is generally
the object in making new and independent determinations to
eliminate as far as possible a new class of errors by bringing
them into the category of accidental errors which tend to
neutralize each other in the final result. The probable error
of the result cannot now be derived from the given probable
errors, but must be inferred from the determinations themselves
considered as observed values, because we now take cognizance
of errors which are not indicated by the given probable errors.
104. When there are but two observed values, formula (4),
Art. 72, becomes
§ VII.] DISCORDANT DE TERMINA TIONS. 83
in which p^ , p^ are the weights assigned to the two values.
Denoting the difference by d, the residuals have opposite signs,
and their absolute values are
Pi +Pi pi + pa
Substituting these values, we have for the probable error of the
mean
P1+P2 pi + pi '
When/, = /a. this becomes
^0= -^ = 0.3272 d. (2)
In the example given in Art. 102, the value of i?o thus obtained
is o".742, which, owing to the discordance of the two given
determinations, considerably exceeds each of the given probable
errors.
Of course no great confidence can be placed in the results
given by the formulae above on account of the small value ofn.*
105. Since the error of each determination is the sum of its
accidental and systematic error, if .Si and s^ denote the probable
*The argument by which it is shown that the value of A deduced in
Art. 69 is the most probable value involves the assumption that before
the observations were made all values of A are to be regarded as equally
probable ; just as that by which it is shown that the arithmetical mean
is the most probable value of the observed quantity a involves the assump-
tion that before the observations all values of a were equally probable. In
the case of a, the assumption is admissible with respect to all values of a
which can possibly come in question. But, in the case of A, this is not true ;
because (supposing « = 2 as above) when d = o the value of A is infinite,
and when d is small the corresponding values of A are very large, so that
it is impossible to admit that all values of A which can arise are a priori
equally probable.
In the present application of the formula, however, these inadmissible
values do not arise, because we do not use it when d is small, employing
instead the method of Art. 99 and the formula of Art. loi.
84 INDEPENDENT DETERMINATIONS. [Art. 105
systematic errors, the probable errors of the two determinations
when both classes of errors are considered are
The proper ratio of weights with which the determinations
should be combined is F\ : R\. The method of procedure
followed in Art. 99 assumes that Sx and s^ vanish. On the other
hand, in the process employed in Art. 102 we are guided, in an
assumption of the ratio R\ : FX, by a consideration of the value
which the ratio s\ : s{ ought to have.
For example, in the illustration, Art. 102, the ratio R\ : J?l is
taken to be one of equality, whereas the hypothesis we desired
to make was that Si=: Si, so that we ought to have
Rl-Rl = ^,-rl.
On the hypothesis Ri = R^ the value of each of these prob-
able errors is, in accordance with equation (2), Art. 104, pd. In
the example this is i".05. If we take (1.05)' as the average
value of Rl and Rl , and introduce the condition written above,
we shall find as a second approximation to the value of the ratio
Rl: Rl about 15:13. The final value corresponding to this
ratio of weights is 14° 53' 13".!, and its probable error as deter-
mined by equation (i). Art. 104, is slightly less than that before
found, namely, Ro — o".740.
Indicated and Concealed Portions of the Risk of Error,
106. It will be convenient in the following articles to speak
of the square of the probable error as the measure of the risk
of error.
The foregoing discussion shows that the total risk of error,
R', of any determination consists of two parts, r" and i", of
which the first only is indicated by discrepancies among the
observations of which the given determination is the mean. It
is only this first part that can be diminished by increasing the
number of the constituent observations. The remaining part
remains concealed, and cannot be diminished until some varia-
§ VII.] PORTIONS OF THK TOTAL RISK OF ERROR. 85
tion is made in the circumstances under which the observationsi
are made, giving rise to new determinations. When the indi-
cated portions of the risk of error in the several determinations
are sufficiently diminished, discordance between them must
always be expected, and this discordance brings into evidence
a new portion, but still it may be only a portion, of the hitherto
concealed part of the risk of error.
107. What we have called in Art. 103 discordant determina-
tions are those in which the indication of this new portion of
the risk of error, to which corresponds the relative systematic
error, is unmistakable, because of its magnitude in comparison
with what remains of the portion first indicated in the separate
determinations, that is, r\ and r\. On the other hand, the con-
cordant determinations of Art. loi are those in which the new
portion is so small compared with ri and 7% as to remain con-
cealed.
Thus, to return to the illustration discussed in Art. 99, if
vwenty times as many observations had been involved in the
determination by the transit, its probable error would have
been reduced to equality with that of the determination by the
theodolite. But if this had been done we should almost cer-
tainly have found the determinations discordant ; that is to say,
the ratio in which the difference between the determinations is
reduced would be much less than that in which the probable
relative accidental error V {A + 'I) is diminished. The ratio in
which the remaining difference between the determinations
should be divided in making the final determination now
depends upon our estimate of the comparative freedom of the
instruments from systematic error,* but the important thing to
be noted is that the probable error of the result would now be
found as in Art. 104, and would be greater than those of the
*It may be assumed that, when the instruments are carefully adjusted,
the one which is less liable to accidental errors is correspondingly less
liable to systematic errors. But this comparison is concerned with the
probable errors of a single observation in each case, and not with those of
the determinations themselves.
86 /NDEPENDENT DETERMINATIONS. [Art, 107
separate determinations. Thus the apparent risk of error would
be increased by making a new determination, but this is only
because a greater part of the total risk of error has been made
apparent, and the result is so much the more trustworthy as a
greater variety has been introduced into the methods employed.
The Total Probable Error of a Determination.
108. In the illustrations given in Arts. 99 and 102 it was sup-
posed that two determinations only were made, so that we had
but a single discrepancy upon which to base our judgment of the
probable amount ofthe relative systematic error. But, in general,
what are regarded as determinations at one stage ofthe process
are at the next stage treated as observations which may be
repeated indefinitely before being combined into a new deter-
mination. Let one of the determinations first made be the
mean of n observations equally good, and let r be the probable
error of a single observation. Then the probable accidental
error of the mean is r^ = -t— • Now, if R is the probable error
of the final value as obtained directly from the discrepancies
of the several determinations, (their number being supposed
great enough to allow us to obtain a trustworthy value,) we shall
find that R exceeds r^, and putting
^' = J + ^ (0
r\ is the new portion of the risk of error brought out by the
comparison of the determinations.
r^
109. The form of this equation shows that when — is already
small compared with r\, the advantage gained by increasing thfe
value of n soon becomes inappreciable.
For example, the reticule of a meridian circle is provided
with a number of threads, in order that several observations of
time may be taken at a single transit. If seven equidistant threads
are used, the mean of the times is equivalent to a determination
§ VII.] THE TOTAL PROBABLE ERROR. S}
based upon seven observations of the time of transit. Chauvenel
found that, for moderately skilful observers, the probable acci-
dental error of the transit over a single thread of an equatorial
star isr — o=.o8, whence for the mean of the seven threads we
have ro = o'.03. The probable error of a single determination
of the right ascension of an equatorial star was found to be
J? = o'.o6, so that, from Ji^ = rl+ rf we have rj = o'.052. The
conclusion is reached that " an increase of the number of threads
would be attended by no important advantage," and it is stated
that Bessel thought five threads sufficient.*
110. Suppose the value of Ji' in equation (i), Art. io8, to
have been derived from the discrepancies of n' determinations of
equal weight. A systematic error may exist for these n'
determinations, and j, being its probable value, we shall have
that is to say, the concealed portion of the risk of error in one
of the original determinations has been decomposed into two
parts, one of which has been disclosed at the second stage of the
process, while the other remains concealed.
The total risk of error in a single one of the n' determina-
tions is R^ + sf, and that of the mean of the determinations is
w
In like manner, if at a further stage of the process we have the
means of finding the value of the probable error R^ of this new
determination by direct comparison with other coordinate deter-
minations, a portion of the value of jj will be disclosed, and we
shall have
n nn n'
where again it must be supposed that a portion s\ of the risk of
error still remains concealed.
* Chauvenet's " Spherical and Practical Astronomy," vol. ii, p. 194
et seq.
88 INDEPENDENT DETERMINATIONS. [Art. iii
111. The comparative amounts of the risk of error which are
disclosed at the various stages of the process depend upon the
amount of variety introduced into the method of observing.
Thus, to resume the illustration given in Art. 109, if the star
be observed at n' culminations, r^ will correspond to errors
peculiar to a thread, and r\ will correspond to errors peculiar to a
culmination. Again, if different stars whose right ascensions are
known are observed, in order to obtain the local sidereal time
used in a determination of the longitude, r\ will correspond to
errors peculiar to a star, together with instrumental errors
peculiar to the meridian altitude.
The Ultimate Limit of Accuracy.
112. The considerations adduced in the preceding articles
seem to point to the conclusion that there must always be a
residuum of the risk of error that has not yet been reached, and
thus to explain the apparent existence " of an ultimate limit of
accuracy beyond which no mass of accumulated observations
can ever penetrate."* But it does not appear to be necessary
to suppose, as done by Professor Peirce, that there is an absolute
fixed limit of accuracy, due to " a failure of the law of error
embodied in the method of Least Squares, when it is extended
to minute errors." He says: "In approaching the ultimate
limit of accuracy, the probable error ceases to diminish propor-
tionally to the increase of the number of observations, so that
the accuracy of the mean of several determinations does not
surpass that of the single determinations as much as it should
do, in conformity with the law of least squares ; thus it appears
that the probable error of the mean of the determinations of the
longitude of the Harvard Observatory, deduced from the moon-
culminating observations of 1845, 1846, and 1847, is i°.28 instead
of I'.oo, to which it should have been reduced conformably to
the accuracy of the separate determinations of those years."
* Prof. Benjamin Peirce, U. S. Coast Survey Report for 1854, Appendix,
p. 109.
§ VII.] EXAMPLES. 80
To account for the fact cited on the principles laid down
above, it is only necessary to suppose that there are causes of
error which have varied from year to year ; and, recognizing this
fact, we ought to obtain our final determination by comparing
the determinations of a number of years, and not by combining
into one result the whole mass of observations.
Examples.
1. In a system of observations equally good, r being the
probable error of «* single observation, if two observations are
selected at random, what quantity is their difference as likely as
not to exceed ? rsj 2.
2. In example i, what is the probability that the difference
shall be less than r? O.367.
3. When two determinations are made by the same method,
show that the odds are in favor of a difference less than the sum
of the two probable errors, and against a difference less than the
greater of the two, and find the extreme values of these odds.
66 : 34 and 63 : 37.
4. A and B observe the same angle repeatedly with the same
instrument, with the following results :
A
B
47° 23' 40"
47° 23' 30^'
47 23 45
47 23 40
47 23 30
47 23 50
47 23 35
47 24 00
47 23 40
47 23 20
Show that there is no evidence of relative systematic (personal)
error. Find the relative weights of an observation by A and
by B, and the final determination of the angle.
100: 13; 47° 23' 38".23± i".62.
5. Show that the probable error in example 4 as computed
from the ten observations taken with their proper weights is
i".53, but that derived from the formula of Art. 104 is o".43,
which is much too small. (See foot-note, p. 83.)
go INDEPENDENT DETERMINATIONS. [Art. Iia
6. Two determinations of the length of a line in feet give
respectively 683.4 ±0.3 and 684.9 i o-3i there being no reason
for preferring one of the corresponding precise values to the
other ; show that the probable error of each of the precise values
(that is, the systematic error of each determination) is 0.65 ; and
that the best final determination is 684.15 ± 0.51.
7. Show generally that when the weights are inversely pro-
portional to the squares of the probable errors, the formula of
Art. 104 gives a value of R greater or less than that given by
the formula of Art. loi, according as d is greater or less than
the relative mean error.
VIII.
Indirect Observations.
Observation Equations.
113. We have considered the case in which a quantity
whose value is to be determined is directly observed, or is
expressed as a function of quantities directly observed. We
come now to that in which the quantity sought is one of a
number of unknown quantities of which those directly observed
are functions. The equation expressing that a known function
of several unknown quantities has a certain observed value is
called an observation equation. Let /» denote the number of
unknown quantities concerned. Then, in order to determine
them, we must have at least [i. independent equations. Thus,
if two of the equations express observed values of the same
function of the unknown quantities, they will either be ident-
ical, so that we have in effect only /*— i equations, or else they
will be inconsistent, so that the values of the unknown quan-
tities will be impossible. So also it must not be possible to
derive any one of the /i equations, or one differing from it only
in the absolute term, from two or more of the other equations.
114. If we have no more than the necessary /i equations, we
shall have no indication of the precision with which the obser-
vations have been made, nor, consequently, any measure of the
precision with which the unknown quantities have been deter-
mined. With respect to them, we are in the same condition as
when a single observed value is given in the case of direct
observations.
Now let other observation equations be given, that is tb say,
let the values of other functions* of the unknown quantities be
observed. The results of substituting the values of the unknown
*Itisnot necessary that these additional equations should be inde-
pendent of the original y. equations, for an equation expressing a new
observed value of a function already observed will be useful in deter-
mining the precision of the observations.
92 INDIRECT OBSERVATIONS. [Art. 114
quantities will, owing to the errors of observation, be found to
differ from the observed values, and the discrepancies will give
an indication of the precision of the observations, just as the dis-
crepancies between observed values of the same quantity do, in
the case of direct observations.
115. As an example, let us take the following four observa-
tion equations* involving x,y and z:
X— y+ 22= 3,
3X + 2y-52= 5,
4;tr + J/ + 4z= 21,
— x + 3j/ + 2Z=H.
jf we solve the first three equations we shall find
x=2^, y = z\, z=\\.
Substituting these values in the fourth equation, the value of
the first member is i2f, whereas the observed value is 14; the
discrepancy is \\. If the values above were the true values,
the errors of observation committed must have been o, o, o, i-f ;
but, since each of the observed quantities is liable to error, this
is not a likely system of errors to have been committed. In
fact, any system of values we may assign to x,y and z implies
a system of errors in the observed quantities, and the most
probable system of values is that to which corresponds the
most probable system of errors.
116. In general, let there be m, observation equations,
involving /t unknown quantities, trO'ix; then we have first to
consider the mode of deriving from them the most probable
values of the unknown quantities. The system of errors in the
observed quantities which this system of values implies will
then enable us to measure the precision of the observations.
Finally^ regarding the /i unknown quantities as functions of the
m observed quantities, we shall obtain for each unknown quan-
tity a measure of the precision with which it has been deter-
mined.
• Gauss, " Theoria Motus Corporum Coelestium," Art. 184.
§ VIII.] REDUCTION TO LINEAR FORM. 93
The Reduction of Observation Equations to the Linear Form.
1 17* The method of obtaining the values of the unknown
quantities, to which we proceed, requires that the observation
equations should be linear. When this is not the case, it is
necessary to employ approximately equivalent linear equations,
which are obtained in the following manner.
Let X, V, Z, . . . be the unknown quantities, and Mi,
M2, . . . Mm the observed quantities; the observation equations
are thee of the form
MX,V,Z,...:) = Mi,
/,{X, Y,Z,...) = M„
MX,Y,Z,...) = M^,
where /i ,/i, . . ./m are known functions. Let Xa, Vo, Zo, . ..
be approximate values of X, V, Z, . . ., which, if not otherwise
known, may be found by solving a of the equations ; and put
X=Xo + x, Y=Y„+y, ....
so that j:,jv, 2', . • . are small corrections to be applied to the
approximate values. Then the first observation equation may
be written
/I (^o + X, Yo+y,Zo + z,...) = Mi,
or, expanding by Taylor's theorem,
where the coefficients 01 x,y,z,... are the values which the
partial derivatives oi fii^X, Y, Z,. ..) assume when X=Xa,
Y= Yo, Z= Zo, • ", and the powers and products of the
small quantities x,y,z,.,.are neglected as in Art. 91.
Denoting the coefficients of x, y, z, . . . hy ai, 61, Ci, . .,
putting «i for M -/i (.Xo, Vo, Zo,.. ), and treating the othet
observation equations in the same way, we may write
94 INDIRECT OBSERVATIONS. [Art 117
a^x + b^y + c^s + , .. . = m^
(I)
am^+ 6my+ CmZ+ ...=»»,■
for the observation equations in their linear form-
118. Even when the original observation equations are in the
linear form, it is generally best to transform them as above, so
that the values of the unknown quantities shall be small.
Another transformation sometimes made consists in replacing
one of the unknown quantities by a fixed multiple of it. For
example, if the values of the coefficients oiy are inconveniently
large they may be reduced in value by substituting ky for_y
and giving to >^ a suitably small value.
119. In the observation equations (i), the second members
may be regarded as the observed quantities, since they have the
same errors. If the true values o{x,y, z , = , . are substituted in
these equations they will not be satisfied, because each n differs
from its proper value by the error of observation v; we may
therefore write the equations
OiX + b^y + t iS' + . , . — Ml = z^i
(h?c + b^y + ^Tj^' .+ ...— ^2 = f 2
K^)
in which, ifx,y, z, . , . are the true values, v, .Vi,.... Vm are the
true errors of observation, and if any set of values be given to
x,y, z, . ^ ., the second members are the corresponding restd-
uals. These corrected observation equations may be called, the
residual equations.
Observation Equations of Equal Precision.
120. Let us first suppose that the m observations are equally
good, and let h be their common measure of precision. Then,
since v is the error, not only of the absolute term n-^ in the first
of equations (2), but of the first observed quantity Mi,, the prob-
§Vm.] EQUATIONS OF EQUAL PRECISION. 95
ability before the observations are made that the first observed
value shall be M^ is
where, as in Art. 35, Av is the least count of the instrument.
Hence we have, for the probability before the observations are
made that the m actual observed values shall occur,
exactly as in Art. 41. The values oiv\,v\, . . .vl^ being given
by equations (2), this value of /* is a function of the several
unknown quantities ; hence it follows, as in Art. 41, that for any
one of them that value is, after the observations have been
made, most probable which assigns to P its maximum value ;
in other words, that value which makes
^1 + ^a + • • • + ^m = ^ minimum.
Thus the principle of Least Squares applies to indirect as
well as to direct observations.
121. To determine the most probable value of ^, we have, by
differentiation with respect to x,
dvi , «foj , , ^, dvm ^
or, since, from equations (2), Art. i IQ;
dvi _ dvt _ <^^n _ -
^-''" ^-*" ••• dx -^'
a^Vi + a^Vi + . . + OrnVm = O. . . . . (l)
This is called the normal equation/or x. Whatever values
are assigned to y, 2, . . .,'\\. gives the rule for determining the
value of X which is most probable on the hypothesis that the
values assigned to the other unknown quantities are correct.
Since v^,v.i, . . .Vm represent the first members of the obser-
96 INDIRECT OBSERVATIONS. [Art. 121
vation equations (i), Art. 117, when so written that the second
member is zero, we see that the normal equation for x may be
formed by multiplying each observation equation by the coeffi-
cient of X in it, and adding the results.
122. The rule just given for forming the normal equation
shows it to be a linear combination of the observation equations,
and the reason why the multipliers should be as stated may be
further explained as follows : If we suppose fixed values given
\.o y,z,. . . , each observation equation may be written in the
form ax^= N, where N only differs from the observed value
^ by a fixed quantity, and therefore has the same probable
error. Now, writing the observation equations in the form
x =
«1
= jr.,
X —
a.
= Xi,
x =
N^
— -^nij
we may regard them as expressing direct observations oix. 11
r is the common probable error of A'l, ^, . . . .A^, that of
-A^ r . r
— or Xi is — ; that of Xj is — , and so on. Thus the equations
are not of equal precision for determining x, and their weights
when written as above (being inversely as the squares of the
probable errors) are as a? : «2 : . . . : «Ji. It follows that the
equation for finding x is, as in the case of the weighted arith-
metical mean (see Art. 66), the result of adding the above
equations multiplied respectively by af, a|, . . . a^;* that is to
say, it is the result of adding the original observation equations
of the form ax — N^ o multiplied respectively by «! , «2 , . . . ««,.
•It must not be assumed that the weight of the value of x, determined
from the several normal equations, is 2a°, that of an observation being
unity. This is its weight only upon the supposition that the absolute
values of the other quantities are known.
§vin.]
THE NORMAL EQUATIONS.
97
123.
The Normal Equations.
In like manner, for each of the other unknown quantities
we can form a normal equation, and we thus have a system o<
equations whose number is equal to that of the unknown quan-
tities. The solution of this system of normal equations gives
the most probable values of the unknown quantities. Let us
take for example the four observation equations given in Art.
115. Forming the normal equations by the rule given above,
we have
2'jx + 6jf = 88,
6x + ly + e = 70,
jy + 5^2= 107.
The solution of this system of equations gives for the most
probable values,
49154 _
19899
2617 _
737
6633
124. Writing the observation equations in their general form,
UiX + b^y + ... + IJ—ni
a^x + 6iy +...+ht—ni
X —
y =
= 2.47.
= 3-55.
1.92.
(0
amX + 6my + . . . + lmi= nm .
we obtain for the normal equations in their general form,
Id? .x+ lab.y + . . . + Ial.t= Ian
Iab.x+ lb'' .y+ ... + Ibl. t=Ibn I . , (2)
lal.x^ Ibl.y-^ ... -^ 11" .t=^Iln
It will be noticed that the coefficient of the rth unknown
quantity in the Jth equation is the same as that of the jth
unknown quantity in the rth equation; in other words, the
98 INDIRECT OBSERVATJONS. [Art. 124
determinant of the coefficients of the unknown quantities in
equations (2) is a symmetrical one.
Observation Equations of Unequal Precision.
125. When the observations are not equally good, if
^1 ) ^21 • • • ^m
are the measures of precision of the observed values
M^, M^, . . . M^,
the expression to be made a minimum is
h\v\ + h\v\ + . . . + KA,
as in Art. 65. Thus, as in the case of direct observations, if the
error of each observation be multiplied by its measure of pre-
cision so as to reduce the errors to the same relative value, it is
necessary that the sum of the squares of the reduced errors
should be a minimum.
Since z/j = o, »a = o, . . . z^m = o are equivalent to the observa-
tion equations, it follows that, if we multiply each observation
equation by its measure of precision (so that it takes the form
hv = o), we may regard the results as equations of equal pre-
cision.
126. The result may be otherwise expressed by using num-
bers /i,/^ , . . .pm proportional, as in Art. 66, to the squares of
the measures of precision ; the quantity to be made a minimum
then is
p,v\ +piu\ + . . . +/m<,
and the normal equation for x is
A^i^i + pia^V^ -I- . . . + pmamVm = O.
The numbers pi, pi, . . . pm are called the weights of the
observation equations; thus, in the case of weighted equations,
the normal equation for x may be formed by multiplying each
observation equation by the coefficient of x in it, and also by its
weight, and adding the results.
§Vni.] EQUA7V0NS OF UNEQUAL PRECISION. 99
The general form of the normal equations is now
Ipa" . X + Ipab .J/ + . . . + Ipal.t = Ipan
Ipab . X + ipb'' .y + ... -\- Ipbl . t - ipbn
Ipal .x+ Ipbl .;)/+...+ Ip? . t - Ipin
(3)
The result is evidently the same as if each observation equation
had been first multiplied by the square root of its weight, by
which means it would be reduced to the weight unity, and the
system would take the form (2), Art. 124.
Formation of the Normal Equations.
127. When the normal equations are calculated by means of
their general form, a table of squares is useful not only in cal-
culating the coefficients Ipa^, Ipb'' ,.. . IpP, but also in the
case of those of the form Ipab, Ipac, . . . Epan, . . . For,
Since
ab= i[(a + by - a' - b'"'],
we have
Spab = illpCa + by - Ipa' - Ipb';\,
by means of which ^pab is expressed in terms of squares,* Or
for the same purpose we may use
ipab = hilpa' + Ipb" - lp{a - bJI.
In performing the work it is convenient to arrange the coeffi-
cients in a tabular form in the order in which they 'occur in the
observation equations, and, adding a column containing the sums
of the coefficients in each equation, thus,
^1 = dtj -t- 3i -F ... + /, + Ml, etc.,
• If 'Zpab alone were to be found, the formula
Lpab = \ [2/(a -1- bf—1p{a—b)^l ,
derived from that of quarter-squares, would be preferable ; but, since
2/0', 2pb^ have also to be calculated, the use of the formula above,
which was suggested by Bessel, involves less additional labor.
lOO
INDIRECT OBSER VA TIONS.
[Art, I2J
to form the quantities "Spas, "Spbs, ... . "Spns in addition to those
which occur in the normal equations. We ought then to find
2pas = Ipa? + Ipab + . . . + Ipan,
Ipbs = lpab+ Ipb" + . . . + Ipbn,
Ipns = Ipan-^Ipbn + . . . + Ipn^,
and. the fulfilment of these conditions is a verification of the
accuracy of the work.
In many cases, the use of logarithms is to be preferred,
especially when the logarithms of the coefficients in the ob»
servation equations are more readily obtained than the values
themselves.
The General Expressions for the Unknown Quantities.
128. In writing general expressions for the most probable
values of the unknown quantities, and in deriving their prob-
able errors, we shall, for simplicity in notation, suppose that the
observation equations have been reduced to the weight unity as
explained in Art. 126, so that they are represented by equations
(i), and the normal equations by equations (2) of Art. 124.
Let D be the symmetrical determinant of the coefficients of
the unknown quantities in the normal equations, thus
Z> =
la^
lab
Sab
lb'
2al
Ibl
lal Ibl ... 11^
let Dx denote the result of replacing the first column by a
column consisting of the second members, Ian, Ibn, . . „ Iln\
and let Dy, Di, . . . Dt be the like results for the remaining
columns. Then
are the general expressions for the unknown quantities.
(1}
§ VIII.] EXPRESSIONS IN DE TERMINANT FORM. lOI
129. Let the value of x when expanded in terms of the
second members of the normal equations be
X = Qilan + Q^Ibn + . . . + Q^Sln.
(2)
Now, in the expansion of the determinant Z?j, in terms of the
elements of its first column, the coefficients of Ian, l'6n, . . . I'ln
are the first minors corresponding to I'a'', lad, . . . la/, in the
determinant Z>.
Denoting the first of these by Z>i , so that
A =
Id' Ibc
Ibc Ic'
Ibl Id
Ibl
Id
II'
it follows, on comparing the values of x in equations (i) and
(2), that
Q.=
Pi
D •
In like manner, the values of Q2, Q^, . . . Q^ are the results of
dividing the other first minors by D.
The Weights of the Unknown Quantities.
130. Let the value oix, when fully expanded in terms of the
second members »i, Kj, . . . «m of the observation equations, be
X = OiWi + a,«j +
+ "m«w
(3)
Then, if rx denotes the probable error of x, and r that of a
standard observation, that is, the common probable error of
each of the observed values Wi, « Wm, we shall have, by
Art. 89,
rl^r'.Ia'.
The precision with which x has been determined is usually
expressed by means of its weight, that of a standard observation
I02 INDIRECT OBSERVATIONS. [Art. 130
being taken as unity. The weights being inversely propor-
tional to the squares of the probable errors, we have, therefore,
for that of X,
131. Since the value- of ;i; is obtained from the normal equa-
tions, we do not actually find the values of the a's ; we therefore
proceed to express I,a? in terms of the quantities which occur
in the normal equations.
Equating the coefficients of «i , «2 , . . . «m in equations (2)
and (3), we find
«i = «ij6*i + hQ-, + . . . + kQ^ \
«2 = ciiQ^ + b^Q^ + . . . + l^Q^
O-m—amQi + bm,Qi+ . . . + ImQp. -
. (I)
Multiplying the first of these equations by cti, the second by
fla , and so on, and adding the results, we have
Sa'^Iaa.Q, + Iba.Q,+ ... + Ila..Q^. . (2)
The value of Saa is found by multiplying the first of equa-
tions (i) by «!, the second by «j, and so on, and adding. The
result is
Saa = la' .Q,+ Iab.Q,+ ... + Sal. Q^. . (3)
Multiplying this equation by D, the second member becomes
the expansion of the determinant D in terms of the elements of
its first column. Hence
Saa = I (4)
In like manner we find
Iba ^Iab.Q, + Ib\Q,+ ... + Ibl. Q^, . (5)
and when this equation is multiplied by D, the second member
is! the expansion of a determinant in which the first two columns
§VIII.] WEIGHTS OF THE UNKNOWN QUANTITIES. I03
are identical. Thus Iba = o, and in the same way we can show
that lea .... iVa vanish.*
Substituting in equation (2), we have now
^«'=5.; (6)
hence from Arts. 130 and 129 we have, for the general expres-
sion for the weight of x,
^'--Q^D. (7)
132. It follows from equation (2), Art. 129, that if in solving
the normal equations we retain the second members in alge-
braic form, putting for them A, B,C, .. ., then ihe weight of x
will be the reciprocal of the coefficient of A in the value of x.'\
In like manner, that oiy will be the reciprocal of the coefficient
of 5 in the value oiy, and so on.
For example, if the normal equations given in Art. 123 are
written in the form
27^ + 6y = A,
6x + i^y + z = B,
j + 542= C,
the solution is
19899;!; = 809^ — 3245 + 6C,
miy = -i2A + 545 - C,
66335' = 2A — gB + 123 C
•Comparing equation (3) with equation (2), Art. 129, we see that Saa
is the value which x would assume if in each normal equation the
second member were equal to the coefficient of x. The system of equa-
tions so formed would evidently be satisfied by ;i; = 1,7 = o, z =; o, , . .
/ = o ; hence Xaa = 1. In like manner, comparing equation (5) with the
same equation, we see that 2*a is the value which x would assume if
the second member of each normal equation were equal to the coefficient
o£^. This value would be zero ; thus 2i5a^ o.
t If the value of the weight of x alone is required, it may be found as
the reciprocal of what the value of * becomes when ^ =: i, B^=.o,
Cz=o, . . . , that is to say, when the second member of the first norma]
equation is replaced by unity, and that of each of the others by zero.
I04 INDIRECT OBSERVA'IIOA'S. [Art. 132
The weights oix,y and z are therefore
A =
809
— 24.60,
A =
737
54
= 13-65,
A =
6633
123
= 53-93-
133. When the value oix is obtained by the method of sub-
stitution, the process may be so arranged that its weight shall be
found at the same time. Let the other unknown quantities be
eliminated successively by means of the other normal equations,
the value ol x being obtained from the first normal equation or
normal equation for x. Then, if this equation has not been
reduced by multiplication or division, the coefficient of A in
the second member will still be unity, and the equation will be
of the form
Rx=T+ A,
where T depends upon the quantities B, C, . . . Now it is
shown in the preceding article that the weight oix is the recip-
rocal of the coefficient of A in the value of x ; hence in the
present form of the equation the weight is the coefficient of ;t;.*
As an illustration, let us find the values of x and its weight
in the example given above, the normal equation being
27^1;+ 6y = 88,
6^ + 15J' + ■^ = 70.
y + 5^2= 107,
The last equation gives
I , 107
2= — —y H '-,
54-^ 54
•The effect of the substitution is always to diminish the coefficient o£
x; for, as mentioned in the foot-note to Art. 122, if the true values of
y, z, , . . t were known, the weight of x would be Sa^, which is the original
coefficient of x, and obviously the weight on this hypothesis would exceed
fx 3 which is the weight when j/, z, . . , I are also subject to error.
§ Vin.] WEIGHTS :.F THE UNKNOWN QUANTITIES. !OS
and if this is substituted in the second, we obtain
■y~ 809 -^ ^ 809 •
Finally, by the substitution of this value of j/ in the first normal
equation, we obtain, before any reduction is made,
19899 „ _ 49154 .
809 809 '
whence
_ 19899 _„j ^-49154
as before found.
The Determination of the Measure of Precision.
134.. The most probable value of^ in the case of observations
of equal weight is that which gives the greatest possible value
to P, Arte 120, that is, to the function
in which the errors are denoted by Mi , Ma , . . » Wm, so that we
may retain Wi , z/j , . . . z'm to denote the residuals which corres-
pond to the values of the unknown quantities derived from the
normal equations. By differentiation we derive, as in Art. 69,
for the determination of h,
J«'=^,. = ..... . (i)
The value of I'm' cannot, of course, be obtained, but it is
known to exceed 2V, which is its minimum value, and the best
value we can adopt is found by adding to Iv" the mean value
of the excess, J«° — Iv*.
135. Let the true values of the unknown quantities be
X ■\- &x,y-\- Sy,.. .t-\- U, while x, y,...t denote the values
derived from the normal equations. We have then the residual
equations
I06 INDIRECT OBSERVATIONS. [Art. IJ5
a^X + b^y + ... + /i/ — «i = !7i
flaX + ^jj/ + . . . + 4/ — «2 = Z/a
O-wP^ "T ^w,y ~r • • • T" ^m^ ^m — — ^w
and, for the true errors, the expressions,
a^l^x + tx') + iJiCjj' + ^j)/) + . . . + /i(i? f 3^) -n^ = u^
a^x + te) + 3j(jj/ + 5j)/) + + 4(/ + Si] -■n^ = u^
OmCx+Sx') + im(y+Sy) + . + 4(/+ 30 — 7Zm= M,„
(i3
..(2)
Multiplying equations (i) by ^i, !72, . . . z'm respectively, and
adding, the coefficient oix in the result is
ChVi + a^V^ + . . . + OmVm,
which vanishes by the first normal equation (i), Art. 121. In
like manner, the coefficient ofy vanishes by the second normal
equation, and so on. Hence
2V = ~ Snv, (3)
Treating equations (2) in the same way, we have
Suv = — 2nv ;
hence
iv'' = i;uv. . . . (4)
Again, multiplying equations (i) by Mi, u^, ^m, and
adding,
Suv = Sau.x + 2bu y + o , . + Slu.i- Inu,-,
and treating equations {2) in the same way.
Sti' = lau (X 4- Sx") + Ibu {y + 3j) + + Slu (f + df) ■ - Inu.
Subtracting the preceding equation, we havj, by equation (4),
Iti' - Iv' ■= lau , Sx + Ibu . 5ji/ + , „ , + IJu M, . (5)
an expression for the correction whose mean value we arn
seeking
§ VIILl THE MEASURE OF PRECISION. IO7
136. Expressions for Sx, Sy, . . . dt are readily obtained as
follows. Treating equations (2) exactly as the residual equa-
tions (i) are treated to form the normal equations, we find
Id" . {x + Sx') + lab . ( J/ + 4r) + ...
+ 2'a/. {t + ^f) = 2an + 2au
lah.ix + Sx') + 2'<5^ (j/ + 5y) + . . .
+ Ibl. (i + df) = Sbn + Ibu
Sal. (x + Sx) + Ibl. (;)/ + 5jj/) 4- . . .
+ 11^ . {t + hf) = Iln + Ilu J
Subtraction of the corresponding normal equation from each
of these gives the system,
Id" . Sx + lab .Sy + . . . + lal . 8t = lau '
lab . dx+ lb' .Sy+ ... + Ibl.St= Ibu
Sal. dx + Sbl . 8y + . . . ■\- Sl\8i = Ilu
a comparison of which with the normal equations shows that
8x, Sy, . . . 8t are the same functions of «i, u^, . . . Um that
x,y, . . .t are of «i, Wj, . . • «m. Hence we have
Sx = O-iUi + a^i + . . . + O-mUm >
where Oi, Oa, . . . an have the same meaning as in Art. 130.
137. Consider now the first term, Sau.Sx, of the value of
Su' — Sv'', equation (5), Art. 135. Multiplying the value of
5;t just found by
SaU ^ fliMi + flSjMj + . . . + aniUm,
the product consists of terms containing squares and products
of the errors. We are concerned only with the mean values of
these terms, in accordance with the law of facility, which is for
each error -7— e~ '''"'. Since the mean value of each error is
zero, it is obvious that the mean value of each product vanishes;
I08 INDIRECT OBSERVATIONS. [Art. 137
SO that the mean value of S,au . te is the mean value of
• Now by Art. 50 the mean value of each of the squares
«f , k| , . . . «^ is — T3 ; hence the mean value of 2au . 8x is —^ ,
or, by equation (4), Art. 131, —p.
In the same manner it can be shown that the mean value of
each term in the second member of equation (5), Art. 135, is
-p; hence that of lu' — Iv' is -^, and the best value we can
adopt for Su' is
Substituting this in equation (i), Art. 134, we have
Iv' = — o— > whence h= J — ^i^.
The Probable Errors of the Observations and Unknown
Quantities.
138. The resulting values of the mean and probable error of
a single observation are
- JL- - I ^^ I \
and the probabFe errors of the unknown quantities are
When the observation equations have not equal weights we
§ VIIL)
PROBABLE ERRORS,
109
may replace lii', which represents the sum of the squares 01
the residuals in the reduced equations, by Ipv', in which the
residuals are derived from the original observation equations.
The formulae (i) and (2) will then give the mean and probable
errors of an observation whose weight is unity.
It will be noticed that when /i = i the formulee reduce to
those given in Art. 72 for the case of one unknown quantity.
139. Instead of calculating the values oiv^,Vi,. . . v,n directly
from the residual equations, and squaring and adding the
results, we may employ the formula for Hv' deduced below.
By equation (3), Art. 135,
Iv' = - Inv.
Now multiplying equations (i) of that article by «i , »fj , . . ,
respectively, and adding the results, we have
Mm
Snv
Therefore
Iv' = In'
lan.x + Idn.y + . . . + Iln.i— In'.
Ian . X •
Ibn.y — ... — Iln . i. . (1)
The quantity In^ which occurs in this formula may be calcu-
lated at the same time with the coefficients in the normal equa-
tions. It enters with them into the check equations of Art. 127.
We may also express Iv' exclusively in terms of these quan-
tities, for if we write
Dn =
and consider the develooment of Dn in terms of the elements
of its last row, we see that
Id'
lab .
. lai
Ian
lad
•
16' .
■
.. I6i
•
I6n
•
•
I'ai
•
Idi .
.. IP
•
Iln
< Ian
Ib7: .
.. Iln
In'
i?„ = — Ian . Dx — Ibn . D«
. — Iht.Di^ In*.D,
I lO INDIkBCT OBSER V A TIJ./VS ^ Ari. ^35
where JD, Da. • Dt have the same meanings as m Art, xsg-
hence
Iv^ = ^. . . . (2)
140. For example, in the case of the four observatiois equa-
tions of Art 115,
X— y -V 2z= 3
3^ + 2j/ - 5^ = 5
4JC + J/ + 4^' = 21
— ■ar + 2U)' + 32=i4J
for which the normal equations are solved in Art 123, the value
of Zr^ is 671 ; and formula (i) gives
l'z^ = 67i -88 X 45154 _7oX 2^
19899 ' 19899
^, 38121 1600
— 107 X ^-5 — = — 5 — ,
' 19899 19899
in which 1600 is the value of Z>„. Substituting this value oi
J'z'" in the formulae of Art 138, we find
e = 0.2836, r = 0.1913
for the mean and probable errors of an observation ; and using
the weights found in Art. 132, we find for those of the unknowi?
quantities
e-8 = 0.057, ^» = 0-077. ^' = 0-039*
rx = 0.038, Ty = 0.052, n = 0.026.
In this example we have found the exact value of I'ti'; if
approximate computations are employed, the formula used has
the disadvantage that a very small quantity is to be found by
means of large positive and negative terms, which considerably
increases the number of significant figures to which the work
must be carried. Thus, because Un' = 671 in the above exam-
ple, the work would have to be carried out with seven-place
logarithms to obtain Hz/' to four decimal places. The direct
§VIII.] MEASURE OF INDEPENDENCE. Ill
computation of the z/^'s from the observation equations would
present the same difficulty in a less degree.
141. Of course, no great confidence can be placed in the
absolute values of the probable errors obtained from so small a
number of observation equations as in the example given above.
There being but one more observation than barely sufficient to
determine values of the unknown quantities, the case is com-
parable to that in which « = 2 when the observations are direct.
By increasing the number of observations we not only obtain
a more trustworthy determination of the probable error of a
single observation, but, what is more important, we increase the
weight, and hence the precision, of the unknown quantities.
The measure in which this takes place depends greatly upon
the character of the equations with respect to independence.
As already mentioned in Art. 113, if there were only /t equa-
tions it would be necessary that they should be independent ;
in other words, the determinant of their coefficients must not
vanish, otherwise the values of the unknown quantities will be
indeterminate. When this state of things is approached the
values are ill-determined, and this is indicated by the small
value of the determinant in question. The same thing is true
of the normal equations. Accordingly, the weights are small
when the determinant D is small ; thus the value of Z) is in a
general way a measure of the efficiency of the system of obser-
vation equations in determining the unknown quantities.
142. If we write the coefficients in the m observation equa-
tions in a rectangular form, thus,
Ox 0>t . . • flfi • • . rX, Y=M^-^y, ... T=M^ + i,
so that x,y,...t are the required corrections to the observed
values. The equations of condition may be reduced as in Art
117 to the linear forms
a^x + a^y + . . . + a^t = E^ '
61X + 6iy + . . . + b^i — Ei
fiX -V/^y + . . . +/^i= E^ .
■ (0
The values oix,y, . . ,t must satisfy these equations, which
are, however, insufficient in number to determine them, and, by
the principle of Least Squares, those values are most probable
which, while satisfying equations (i), make
piX' +piy^ + . . . +/jx^' = a minimum.
In other words, the values must be such that
p^xdx +p.^ydy + .. . + p^id/ = o, ... (2)
for all possible simultaneous values of dx, dy, . . . di, that is, for
all values which satisfy the equations,
a^dx + a^dy + . . . + ay,dt = o '
b^dx + b^dy + . . . + b^dt = o
(3)
/idx + f^dy + . . . + f^dt = o J
derived by differentiating equations (1). Hence, denoting the
§VIII.]
CONDITIONED OBSERVATIONS.
"5
first member of equation (2) by P and those of equations (3)
by ^1, kSi, . . . kSI, , the conditions are fulfilled by values which
satisfy equations (i) and make
■ir ^~ lC\*3\ *~" ^gOg ~" • I
• kvS, = o,
(4)
where ky, k^, . . . kv are any constants.
This last equation will be satisfied if we can equate to zero
the coefficient of each of the differentials, thus putting
piX = kyOy, + AA + . . . + kvfl
P^y = kyfli + kj>i f . . . + kvft
' I
(5)
and this it is possible to do because we have fi unknown quan-
tities and V auxiliary quantities Jbi, k^, . . . kr which can be
determined so as to satisfy the v + /^ equations comprised in
the groups (i) and (5).
148. Substituting the values oix,y, . . . /from equations (5)
in equations (i), we have a set of linear equations to determine
the i's which are called the correlatives of the equations of con-
dition. These equations may be written in the form
P P
hS'^ +k,I^4 + •'■+ *.^-$ = ^
(6)
P ■ "- P ■ ■*' ' """"/
in which the summation refers to the coefficients of the several
unknown quantities ; thus, for example, ^ -^ is the sum of the
squares of all the coefficients in the first equation of condition
each divided by the weight of the corresponding unknown
quantity. The correlatives being found firom these equations,
Il6 INDIRECT OBSERVATIONS. [Art. 148
the values of the corrections x,y,. . .t are given at once by
equations (5).
149. When there is but one equation of condition
OxX + a^y + . .. + af,i=£,
the second members of equations (5) reduce to their first terms,
and the equations require that the corrections of the several
unknown quantities shall be proportional to their coefficients
in the equation of condition divided by their weights. Equa-
tions (6) then reduce to the single equation
k£^ = £,
and the corrections are
•* ^3 ■*^» y ^3 -"i
2"- 2-
P P
In the very common case in which the numerical value oi
each coefficient in the single equation of condition is unity (for
example, when the successive angles at a point, or all the
angles of a polygon, are measured, or when the sum of two
measured angles is independently measured), we have the
simple rule that the corrections are inversely proportional to
the weights.
Examples.
I. Denoting the heights above mean sea level of five points
by X, Y, Z, U, V, observations of difference of level gave, in
feet:
-^= 573-08 Z- 3^=167.33 U- F= 425.00
Y-X= 2.60 U-Z= 3.80 F= 319.91
J^= 575-27 u-Y= l^o.2& y= 319-75
Putting X= 573 + X, y= 575 +y, Z= 742 + z, U= 745 + tt^
§VIII.]^ EXAMPLES. 117
F= 320 + V, find the values and probable errors of the cor-
rections x,y, z, u, V, supposing the observations to have equai,
weight.
J? = — 0.19 ± 0.23, J/ = 0.14 ± 0.21, ^^ = 0.05 ± 0.30..
M= 0.43 ±0.25, 2/ = 0.03 ±0.19.
2. Given the observation equations :
•*^ = 4-5. y=i.6, x—y=2.T,
with weights 10, 5 and 3 respectively, determine the values of
X and J/. X = 4.468 ± 0.049, J' = ^-^^3 ± 0.063.
3. Measurements of the ordinates of a straight line corres-
ponding to the abscissas 4, 6, 8 and 9, gave the values 5, 8, 10
and 12. What is the most probable equation of the line in the
formj/ — mx + i? jy = i-339-^ — 0.029.
4. Given the observation equations of equal weight :
x= 10, y— x=T, jj/ = 18,
y — z=.g, X — z= 2,
determine the most probable values of the unknown quantities,
and the probable errors of an observation and of each unknown
quantity. ;ir=iol, j/=i7l, 2 = 8i,
r = ri= 0.29, rx = ry = 0.23.
5. In order to determine the length x at 0° C. of a meter
bar, and its expansion y for each degree of temperature, it was
measured at temperatures 20°, 40°, 50°, 60°, the corresponding
observed lengths being 1000.22, 1000.65, 1000.90 and 1001.05
mm. respectively. Find the probable values of x and y with
their probable errors. x = 999""".8o4 ± 0.033,
y = o'°°'.02I2 ± 0.0007.
6. The length of the pendulum which beats seconds is known
to vary with the latitude in accordance with Clairant's equation,
/=/' + ^-§-y.-M)/'sin'A
where I' is the length at the equator, }■ the ratio -j^^ of the cea-
.Tl8 INDIRECT OBSERVATIONS. [Ait. 149
trifugal force at the equator to the weight, and ;* the compres-
sion of the meridian regarded as unknown. Putting
/' = 99i- + ^, (l^-;.^/'=_y,
observations in different latitudes gave in millimeters :
X + 0.969J)/ = 5.13, x-\- o.ogsjj' = 0.56, X + o.2,vjy = 1.70,
^ + 0.749JJ' = 3-97» X =0.19, .r + 0.685J)/ = 3.62,
X + 0.426^ = 2.24, ;«: + o. 1 527 = 0.77, X + 0.793J/ = 4.23.
Find the length at the equator with its probable error.
/' = 99i""".o69 ± .026.
7. Find the value of fi in the preceding example and its prob-
able error. /t = ^ ± 0.00046.
8. The measured height in feet of .(4 above O, B above A and
B above O are 12.3, 14. i and 27.0 respectively. Find the
most probable value and the probable error of each of these
differences of level. 12.5 ± 0.17; 14.3 ± 0.17; 26.8 ± 0.17.
9. A round of angles at a station in the U. S. Coast Survey
was observed with weights as follows:
65° 1 1' 52".500 with weight 3, 87° 2' 24".703 with weight 3,
66 24 15 .553 " " 3, 141 21 21 .757 " " i;
find the adjusted values whose sum must be 360°.
65° II' 53".4i45. 87° 2' 25".6i7S,
66 24 16 .4675, 141 21 24 .5005.
10. Four observations on the angle A" of a triangle gave a
mean of 36° 25' 47", two observations on Y gave a mean oi
90° 36' 28" and three on Zgave 52° 57' 57". Find the adjusted
values of the angles and the probable error of a single obser-
vation, r = 7".7 ; A" = 36" 25' 44"-23,
Y= 90 36 22 .46,
-^=52 57 53 'SI'
tl. A round of four angles was observed as follows :
Z^" 52' 14". 28 weight 2, 44" 35' 56".54 weight 3,
145 23 16 .35 " 4, 131 16 21 .47 " 3,
find the adjusted values.
38° 51' 35".94. 44' 35' 30^'-9^
145 22 57 .18, 131 9 55 .9^
§vni.] EXAMPLES. ^'9
12. Measurements of the angles between surrounding stations
were made with weights as follows :
Between stations i and 2, 55° 57' 58".68, weight 3^
" 2 " 3. 48 49 13 -64, " 19,
" " I " 3- 104 47 12 .66, " 17,
" 3 " 4, 54 38 15 .53, " 13.
« « 2 " 4, 103 27 28 .99, " &
Find the corrections of the angles in the order given.
or'.a85, + lbl\y + \cl\z + . . . + \ll\t = [/«] .
As mentioned at the end of Art. 126, we may suppose the
observation equations (i) to have been reduced to the weight
unity, so that \aa\, \ab\ . . . \ln\ stand for ^a", ^ab, . . 2/«.
§ix.]
THE HEDUCED NORMAL EQUATIONS.
121
151. The value of x in terms of the other unknown quanti-
ties derived from the first of equations (2), or normal equation
for x^ is
\aa\ [aa]
.+
\aa\ '
Substituting this in the f^— i other equations, they become
(M-Mg]).+ (M-M[^])
.+.
■ [««] - M
([./]-[./]g-5.+...+(m-M[3)-M-Mg]
in which it will be noticed that the coefficients of the unknown
quantities have the same symmetry as in the normal equa-
tions (2). These equations for the /^ — i unknown quantities
y, z, . . . f are called the reduced normal equations, and are
written in the form
\bb, i]jc + lie, i]0 + . , . + \U, i]/ = \hn, i]
\bc, i]j»' -f- \cc, i> -h . . . + \cl, i]/ = \cn, i]
\bl, i]jc -f U !>+... + [//, i]^ = lln, ij .
(3)
it. which
■ • • • *
{4V
122 GAUSS'S METHOD OF SUBSTITUTION. [Art. 151.
Equations (4) show that the rule for the formation of the
coefficients and the second members of the reduced normal
equations is the same throughout; namely, from the correspond-
ing coefficient in the normal equations we are to subtract the
result of multiplying together the two expressions in whose
symbols one of the letters in the given symbol is associated
with a, and dividing the product by \ad\.
The Elimination Equations.
152. Eliminating y by means of the first of the reduced
normal equations (3) from each of the others, just as x was
eliminated from the normal equations, and employing a similar
notation, we have the fi — 2 equations
\cc, 2]^ + . . . + [cl, 2]/ = [en, 2] \
[. • . (S)
which may be called the second reduced normal equation*.
The coefficients in these equations are derived from those in
equations {3) exactly as the latter were found from those in
equations (2). Thus
(6)
In like manner the third reduced normal equations are
formed from these last, the coefficients being distinguished
by the postfixed numeral 3, corresponding to the number of
§ IX.] THE ELIMINATION EQUATIONS. 123
variables which have been eliminated. We finally arrive at
the single equation
[//, n- x'\t = [In, f^- 1] (7)
which determines the unknown quantity standing last in the
order of elimination.
153. The quantity which immediately precedes / is next
derived from the first of the preceding set of equations (that
is, from the equation by means of which it was eliminated) by
the substitution of the numerical value found for / ; and so
on, until finally x is found from the first of the original normal
equations. The equations from which the unknown quantities
are actually determined are therefore the following :
[aa]x + [a^]j)' + [m]:! + ■ ■ • + [al]t = [an]
[6l>, i]y + [be, !]«+... + [bl, i]/ = [bn, i]
[cc, 2]^ + . . . + [cl, 2]/ = \cn, 2]
(8)
[//,;U-l]/= [/«,/*-!].
These are called the finai, or elimination equations.
The Reduced Observation Equations.
154. Let us suppose that there exists a relation between
the variables which must be exactly satisfied, while the m
observation equations are to be satisfied approximately. Let
this relation be
«* + /ly + . . • + ^' = " (0
Eliminating x from the observation equations (i), Art 150
by the substitution of
x-=—-y — -z—-.. / + -I
a-' a a a.
124
GAUSS'S METHOD OF SUBSTITUTION: [Art. 154.
derived from this equation, we have
{b- a,-jy +[c,- a,^z + . . . + [i,- a,-^t^n,- a^
f^m— ^m-jy+Km— «m^j«+ • • • + f 4 — ««- j ^= «m " ««-
which may be called the reduced observation equations, and
written in the form
b:y + c:z + . . . + /.v = «/ ^
h:y + f/^ + . . . + i:t = «/
bmy + c^z + . . . 4- 4V = nj .
■ . (2)
a comparison of which with the equations written ' above suffi-
ciently indicates the values of b^ , <:/, . . . «„', • • • nj-
The /^ — I normal equations derived from these are
lb'b'-\y + \b'c'\z + . . . + Ib'l'y = Vb'n'\
Ib'c'^y + [.V']^ + . . . + Vnt = [^'«'l
(3)
in which
.(4)
§IX.] THE REDUCED OBSERVATION EQUATIONS. -125
155- Let us now suppose that the equation of condition
(i) which is to be exactly satisfied is identical with the first
of the normal equations (2) of Art 150, so that
a = \ad\, ^=\ab\, ... v=.\an\\
then equations (4) become
\b'b'\ = \bb\ - P^°
\ad\
[,v] = \pc\ - MM
\aa\
\l'n'\ = [/«] -
(S)
Comparison of these with equations (4), Art. 151, shows that
the normal equations (3) of the preceding article now become
identical with the first reduced normal equations of Art. 151.
Hence the first reduced normal equations are the same as the
normal equations corresponding to the reduced observation equations
which would result if x were eliminated from the observation
equations by means of the normal equation for x.
It is evident that, in like manner, the second reduced
normal equations are the same as the fx — 2 normal equation
which would result from the reduced observation equations, if
they were further reduced by the elimination of y by means
of the reduced normal equation for j/ ; or, what is the same
thing, the normal equations which would result if x and j' were
eliminated from the original observation equations by means
of the normal equations for x andj. Similar remarks apply to
the other sets of reduced normal equations.
156. An important consequence of what has just been
proved is that, among the coefficients in the reduced normal
equations, or auxiliary quantities, those of quadratic form,
{bb, ij, \cc, i], ... {cc, 2I, ... [//, // — i],
126 GAUSS'S METHOD OF SOBSTITUTION. [Art. 156.
being, like the corresponding quantities in the normal equa-
tions, sums of squares, are all positive. It is further to be
noticed that each of these quantities decreases as its postfix
increases, for the subtractive quantities in the formation of
the successive values are themselves positive. For example,
Weights of the Two Quantities First Determined.
157. The unknown quantity / has been determined in
equation (7), Art. 152, after the manner described in Art. 133;
that is to say, from its own normal equation — no reduction
by multiplication or division having taken place in the course
of the elimination. Hence, as proved in that article, its weight
is the coefficient of the unknown quantity; that is to say, the
weight of an observation being unity, that of / is
pi = [//, /< - i].
which, as shown in the preceding article, is necessarily a posi-
tive quantity.*
The weight of any one of the unknown quantities might be
determined, in like manner, by making it the last in the order
of elimination.
158. Let s be the unknown quantity preceding t, so that
[//,;.-.] = [//,;,- 3] -Ig^^',
• As shown in Art. 156, the substitutions diminish the successive coef-
Bcients of (. Compare the foot-note to Art. 133, p. 104. In fact [//] is
the weight that t would have if the true values of all the other quantities
were known; [//, i] is the weight which it would have if all the others
except X were known— that is, if x and / were the only quantities subject
to error ; and so on.
§ IX.] THE REDUCED EXPRESSION FOR 'Sxi'. 12J
or
[//, )X - {\lkk, fl-2-]= [II, fl - iMkk, Jil-2]- [kl, M - 2]\
If now the order of s and t be reversed, no other change of
order being made, the auxiliaries with the postfix /i — 2 will
be unaltered, and we shall have
[kk, M - i][//. M-2] = [M, f* - 2][//, /i - 2] - [^l,M- 2]',
hence
[kk,fx - i][//, /^ - 2] = [//, M - i][kk,M - 2].
But [M, ;u — i] is the weight of s, therefore we have
The weights of the other unknown quantities cannot be
thus readily expressed in terms of the auxiliaries occurring
in the calculation of /. A general method of obtaining all the
weights will be given in Arts. 174-176.
TAe Reduced Expression for 2v*.
159. We have found in Art. 139 for 2»' or [vv] the expres-
sion
\vv\ = — [an\x — [bn\y - ... — [/«]/ + [««],
which is similar in form to the expressions equated to zero
in the normal equations. If in this we substitute the value
of X, as in Art. 151, it becomes
[w'\ = — \pn, i\y- [en, i]z— ... - [/«, i]/ + [tm, i],
128 GAUSS'S METHOD OF SUBSTITUTION. [Art. 159^
in which
\a,a\
\nn, I J = \nn\ — S^— >
\aa\
after the analogy of the auxiliary quantities defined in equa-
tions (4), Art. 151. In like manner, by the elimination of
y, \yv\ is reduced to the form
\yv\ = - [en, 2]z- . .. - \ln, 2]i + \nn, 2],
and finally, by the substitution of the value of t, to
[H = \nn, )x\
the postfix fi indicating that all the unknown quantities have
been eliminated.
Substituting in the expressions for the mean and probable
error of an observation. Art. 138, we have
The General Expression for the Sum of the Squares of the
Errors.
160. The following articles contain an investigation* of
the sum of the squares of the errors considered as a function
of the unknown quantities, showing directly that the minimum
* Gauss, " Theoria Motus Corponim Coelestium," Art. 182; Werke
vol. vii. p. 238.
g IX.] SUM OF THE SQUARES OF THE ERRORS. 1 29.
value of this quantity corresponds to the values derived
from the normal equations, and is equal to \nn, ;u], and also
deriving from the general expression the law of facility of
error in /, and thence its weight.
Let
W=[zw] (i)
be the sum of the squares of the errors in the observation
equations, that is to say, of the linear expressions of the form
(Art. 119), •
ax -{- dy -\- . . . -\- /t — n = V.
The absolute term in tV is obviously [««]. Put
Then
X = 'Siy-z- = \av\ = [aa]x + [al>]y + . . . + [a/]f — [an]. (3)
The equations X = o, Y = o, . . . T = o are the normal
equations. Now, since
- -^~- = X-,- = [aa\X, or, - -7- f— -f = X,
2 dx dx 2 dx \aa\
1 d (^.^ X'X
2 dx\ \_aa\l
hence, if we put
W,= W-^-^, (4)
W^ is a function independent of x. Now, in equation (4),
Wi has for all values of the variables which make X =0
130 GAUSS'S METHOD OF SUBSTITUTION. [Art. 160.
the same value as W ; hence W^ is what W becomes when x
is eliminated from it by means of the first normal equation,
X=o.
161. It follows from what has just been proved, that
W, = \y'v'y, (S)
that is to say, W^ is the sum of the squares of expressions of
the form
ey-\-<^z-\- ... +tf-n'=v', '
corresponding to the reduced observation equations, Arts. 154,
155. The absolute term in W^ is therefore [«'«'] or [««, i].
If, now, we put
^•~2 dy' '" ^'~'2 dt' • • • ^^'
Y,=2v'^=[yv']=[yy]y+[6V]z+. . .+[dr]/-[yn']. (7)
ay
and F, = o, . . . T, = o, are the reduced normal equations.
The relation between the expressions y, , . . . y, and
X, Y, . . . 7' is derived from equation (4); thus, differentiating
with respect to y,
K=y-r^^=v-^x,. . . (8)
' [aa] dy [aa]
which gives another proof of the identity of the coefficients
[i'6'], . . . [p'n''\ with \bb, i], . . . \bn, i], established in Art. 153.
We now prove, exactly as in the preceding article, that
is a function independent oiy as well as of x, and is identical
§IX.] SUM OF THE SQUARES OF THE ERRORS. I3I
with \y"v"\ the sum of the squares of expressions of the
form
/'«+...+ /"/ - n" = v"
corresponding to the second reduced observation equations,
from which x and y have been eliminated by means of the
equations ^ = o, K,= o. The absolute term in W^ is obviously
[«"«"] or [««, 2].
162. Proceeding in this way, we finally arrive at an expres-
sion Wy. which is independent of all the variables, and consists
simply of the absolute term [««, /<]. We have thus reduced
W to the form
The denominators \_aa\, \bb, i], . . . [//, /^ — i], being sums of
squares, are all positive; hence the minimum value of JFis the
value \nn, /*] corresponding to the values oi x, y, . . . t which
satisfy the equations -Sf = o, F, = o, . . . 7)i _ , — o.
163. Since W is the sum of the squares of the errors, the
probability that the actual observations should occur is
proportional to ^ -*''*' as in Art. 62. Therefore, by the
principle explained in Art. 30, the observations having
been made, the probabilities of different systems of values of
the unknown quantities are proportional to the corresponding
values of this function. Hence, C being a constant to be
determined, the elementary probability, Art. 21, of a given
system of values ol x,y, . . . t'\%
Ce-'<^^dxdy ...dt, (11)
• This result is also derived by Gauss in a purely algebraic manner in
the " Disquisitio de Elementis EUipticis Paladis; " Wtrke, vol. vi. p. 22.
See also Encke, Birliner Astronomisekes Jahrbuch for 1853, pp. vji-Vll.
132 GAUSS'S METHOD OF SUBSTITUTION. [Art. 163.
where h is the measure of precision of an observation, and C
is such that the integral of the expression for all . possible -
values of the variables is unity.
The probability of a given system of values oi y, z, . . . t,
while X may have any value, is found by summing this
expression for all values of x. It is then
Cdy . ..dt f e-'^'^dx =Cdy...dt e-^'f^'ij'^'^Ux,
since PF, in equation (4) is independent of x. Since — = [aa],
the value of the definite integral in this expression is, by
equation (7), Art. 39,
^'"'^dx = ~ e ^""^ dX =
\aa\ J-"
\aa\ J-" h ^\aa\
Thus the probability of a given system of values of y,
z, . . .t\s,
-P-P^dydz . . . dt e-'^'^^ . . . : . (12)
164. Iri like manner, the probability of a given system of
values oi z . . . i, X and y being indeterminate, is
^ ^'^-''~ . . . di r e-^"^'dy,
h ^[aa]
which, by equations (9) and (7), reduces to
.^^^^^—dz...dte-^"^^. . . (13)
§1X.] PROBABILITY OF A GIVEN VALUE OF t. 133
-^■Proceeding in this way, we have, finally, for the probability
of a given value of t,
. • . .z^"-' \/\\aa\bb, i\. . .\_kk,ti- i'W ■ ■ \ V
Again, integrating this for all values of t, we have
1 -v:: • ^^FTIMPTi] ...[//, /^ - i]} - '• • • ^'5^
Substituting the value of C-thus determined, we obtain foi
the probability of f,
kj\/[n, fl - A^_H^(W^_^-[nn,y.-\)^t. . . . (l6)
^>-- - ■ '=T n .ri',! + [««' l^'\
But
ill, /i - 1]
and , . . .
7; _, = [//, /^ - i\t - {In, /< - i];
therefore, putting
_ T^-r _ _ [ In, /^ - i] ; ; .
afed viaittm^ dt, the expression (16) gives for the law of facility
of'error in /, ^
l;^_,; . •. : AVL^^' y- — ^1 ^ -A»r//. »• --^It' (17)
134 GAUSS'S METHOD OF SUBSTITUTION. [Art. 164.
This is of the same form as the law of facility for an ob-
servation, except that the measure of precision is
h V[U, M - i].
Thus the most probable Value of t is that which makes
T = o, namely,
and the weight of this determination, when that of an observed
quantity is unity, is
/. = U^, it - i].
TAe Auxiliaries Expressed in Determinant Form.
165. If, in the determinant of the coefficients of the normal
equations, denoted by D in Art. 128, we subtract from the
second row the product of the first row multiplied by % — 4, it
becomes
o. [^*f i]i \pi:, i], . . • \pl, i].
Treating the other rows in like manner, the determinant D
is reduced to a form in which the first row is unchanged, and
the rest are replaced by a column of o's and the determinant
of the first reduced normal equations. Denoting this last de-
terminant by Z>', we have D — \ad\D'.
By a similar reduction of 2?', D is further reduced to a form
in which the first two rows are as in that described above, and
the rest are replaced by two columns of o's and the determi-
nant, D", of the second reduced normal equations. Finally,
D is thus reduced to the determinant of the elimination equa-
tions (?), Art. 153.
§IX.] AUXILIARIES IN DETERMINANT FORM. 135
The successive forms of D give the equations
D=-\aci\iy-\aa'\\bb, i]Z?"= . . . = \_aa\bb, {\\cc, 2] . . \ll,)X-i\.
166. If, in the form of D involving Z)""', we take the first r
rows, and then any other row (which will therefore be a row
belonging to Z?**"'), the same reasoning shows that any deter-
minant formed by selecting r + i columns of this rectangular
block is equal to the minor occupying the same position in D.
We can now express any auxiliary, say [a/?, r\ as the quo-
tient of two minors, of the (r + i)th and rth degree respec-
tively, in D. This auxiliary occurs in the form of D just
mentioned. Taking the first r rows and columns together
with the row and column in which the given auxiliary occurs,
we have a determinant whose value is
\aa\bb, i] . . . \yY> r - i][aA r\
because all the elements below the principal diagonal vanish.
But this determinant is equal to that similarly situated in Z>,
and the coefficient of [afi, r\ is equal to the determinant
formed from the first r rows and columns of D. For example,
for [de, 2] we have
[aa]
o
o
and
therefore
[abl M
\bb, i] [be, i]
=
[de, 2]
[ad\ [„ of
Art. 139 is ■ .
Z>„ = \aa\b'b, i] ...[//,/< — \\nn, /^] —' iy(nn, }i\; ■
therefore
\nn, >"] = 2f .
which is the same value that was found for \vv\ on j). 1 10.
Form of the Calculation of the Auxiliaries.
168. In calculating the coefficients which occur in the
elimination equations and the value of \vv\, it is important to
arrange the work in tabular- form,' and to apply frequent
checks to the computation to secure accuracy. In the an-
nexed table,* which is constructed for four unknown quanti-
ties, the first compartment contains the coefficients and second
members of the normal equations together with the value of
[««], which are derived from the observation equations, as
explained in Art. 127. The coefficients are entered bpposit^'
and below the letters in their symbols, those below the diag-
onal line, whose values are the same as those symmetrically
situated above, being omitted; Beneath those In thfe first line
are written their logarithms, which are used 'in comfjutlng the
sub tractive quantities placed beneath each of the other por,
efficients. •
* The tabular arrangement is taken from W. Jordan's " Handbuch dett
Vermessungskunde." .See also Oppolzer's "Lehrbuch zur Bahnbestim-
mung der Kometen und Planeten," vol. ii. p. 340 e( seq., where the table,
with a somewhat different arrangeme:nt, is given for six unknown quantities,
and an example is fully worked out. '
§IX.] CALCULATION OF THE AUXILIARIES.
137
a
b
f
a'
«
s
a
\aa-\
log [oa]
\_ab-\
log [fl^]
log [o„K 3]
^
/ =
M =
[»», 4]
[ns, 4]
138 GAUSS'S METHOD OF SUBSTITUTION. [Art. 168.
In expressing the subtractive quantities we have adopted
for abridgment the notation
^'~M' °~\aaY ''~[aay "*""[««]•
The logarithms of these quantities are placed at the side,
and, adding them successively to the logarithms above, the
antilogarithms of the sums are entered in their places. After
this is done, the results of subtraction are the auxiliaries with
postfix I , which are to be placed in corresponding positions
in the compartment below.
In like manner the third compartment is formed from the
second, and in expressing the subtractive quantities we have
put
So also we have put
and finally,
_ [cd, 2] _ [en, 2] .
[cc, 2] [cc, 2]
n _ ['^«. 3 ]
which is also the value of /. Thus the first four compartments
correspond to the several sets of normal equations, and their
first lines to the four elimination equations. Finally, in the
fifth compartment we have computed \nn, 4], which is the
value of [vv].
Check Equations.
169. The column headed j is added for the sake of the
check equations
[a«] + lab'l + M + W\ + [««] + [«^] = o
\_ab\ + \bb-\ + \bc-\ + \bd-\ + \bn\ + \bs-\ = o
[a«] + \bn'\ + [„a,, ... (2)
and, for the determination of the a's,
^„+^,af. + «, =oX. . . . (3)
Aa + BiPi^ + Qa, + «, = o J
In like manner, to find y we multiply the second, third and
fourth of equations (i) by i, A,/?,, respectively, and add.
The result is
y = ^„+C„/J, + ^»/?. (4)
where the /?'s are determined by
Again, multiplying the last two of equations (i) by i, y^,, and
adding
s=C„ + Ar.. (6)
where Yi is determined by
Ci-\rY, = o (7)
173. The form for the computation of «„ a„ a, , fi„ /?„
y„ according to equations (3), (5), and (7), and the numerical
work for the example of Art. 170, is as follows:
.144
GAC/SS'S METHOD OF SUBSTITUTION. [Art. 173.
--^6
-^0
log \
"2
logOfg
logOTj
.050133
.009065
.002146^
— .000020
— .002948
9„26573
.059198
8.77331
— .000822
M1487
-B_
log/3.
8^69164
-B.
d
log'/J.
.000106
.002448^
.002554
7.40722
log Xs = 8^69720
The values of a^, yS^ and ^3, are not found, as their: loga-
rithms only are needed. , ■
We may now recompute the values of the unknown quan^
titles by means of equations (2), (4), (6) by way of verifying
the values of ot^, . . . y,a.s well as those of *, ... A The form
of computation will be as below u . ...,,.
Bn
^»
X
y
z
t
■50325 '
.07927
.00106
. 00000
— .42989
— .00088
— .00001
.017847
.0001229
-.004415
.58358
-.43078
.018067
-.004415
" The numerical values agree exactly with those, fpund in;
Art. 171.
^IXil JVE/G//TS OF THE UNKNOWN QUANTITIES. 145^
The Weights of the Unknown Quantities.
174- Tlie principle' by which we obtain expressions for the
weights is that proved in Art. 132, namely: When the value of
any one of the unknown quantities is expressed in terms of
the second members of the normal equations, its weight is the
reciprocal of the coefficient of the second member of its own
normal equation; or'what^ is the same thing: The reciprocal of
the weight is what the value of the unknown quantity becomes when
the second member of its own normal equation is replaced by unity
and that of each of the others by zero.
Restoring the values of the quantities An, -Bn, Cn, JDn, the
values of x,y, z, t, Art. 172, are
[aa] + [bb, i]!"' '^ ice, 2f ^"^ [dd, 3] '
[a«]
\aa
{bn,
y - \bb, i]
_ \cn, 2]
i] {en, 2 ]^ {dn,i\ ^
+ [.., 2f-+[^^.3r
{cc, 2] + {dd, zf>
{dn, 3 ]
{dd, 3]
W
Equations (3), (5), atid (7), Art. 172, show that the values of
OTj,...;/, are independent of the values of {an\, {bn\, {cri\,
and {dn\ ; hence the changes indicated abovcj in order to con-
vert the second members of equations (i) into ;the expressions^
for the reciprocals of the weights, have only to be made in the
numerators. {ali\, {bh, i]]. \cn, 2],, and , [], p. 140) we find for the mean errors
e, = .02669, £, = .02750, €, = .02283, ^« = -o»»77;
and hence for the probable errors
r, = .01800, r, = .01855, '• = •°^ii9> U = -01536.
r^&
GAUSS':S METHOD OF SUBST:TUTI0.N.. {Art. ifi':
.. .
1
log a^^
logo-/./
log A'
■'"■ -J
log a,'
log ^3'
log r"
^°^ M
[H . ..
'■'
«,' •
I
I
.,[^^, i] -
[M l] ■
'°g[5^,iT
■' ...'■
<'
/S,"
1
I
[^^, 2]
[^^,2]
^-°^ L-,::2l
[-^^.2]
[rf^, 3]
[dd, 3]
Vdd, 3]
'°' K^3i
"'■ ^
- I
I
I
I
A
A
A
A---
log-^
A
A
'!'^
8.53146,
-..;...
7.'S4462 '
■7.38328^
, 3-.82974
4.81444
7 -3944°.
^ .«
-32034
' '
9-50561
:.
.01201
■35318
9.54800
:."j.!. ..;
r-. .
.o'oo8S!
... .-906^9 ,
: -24315
. 9-38587
.OOOBQ
, ,P0QQCS .
_■ ;.QSQ$p ;
9.38483.
f_-
• i\
•SSS'S'o
•353y7- '
•243-75-=
1
:3
9.52270
9.54872 ^
9.38694
9-38483
:■:.'-; ! .-
g IX.] EXAMPLES. 149
Examples.
1. Show that the values of p^ when there are four unknown
quantities given in Arts. 158 and 176 are identical.
2 . Show that the weight of the determination of [6m] is
[M]~^ that of \bn, i] is [W, i]~i, and so on.
3. Show that, if the normal equation for x were known to
be exactly true, the values of the unknown quantities and the
weights relatively to that of an observation of all except x
would be unchanged, and that the weight of an observation
would be increased in the ratio m — }x-\- x : m — y..
4 Solve the following normal equations which resulted from
twelve observation equations :
S.1143X — 0.2792JV + 3.34602 = — 0.7365,
— 0.2792;*: + I4.6i42_j' -|- 0.19582 = 2.1609,
3.3460* + 0.1958;/ + 7.67542 = — 0.8927,
M = 0-5379.
and find the probable errors of the unknjj^wn quantities.
X—- ,0803, y - .1475, z = - .0851;
Tx = .034, Tp = .017, fa = .028.
5. Solve the normal equations '■
5.2485* - 1.74727 - 2-1954^ = — o.-5399>
— 1.7472.J' + I-8859J)' + 0.80412 = 1.4493.
— 2.1954^ + 0.8041;/+ 4.04402 = .1.8681,
[««] = 2.6322;
and given m = 10, find the probable errors.
[tw] = 0.5504, X = 0.422, y = 0.945, z = 0.503;
y = 0.189,. ^a, = 0.108, /-„ = 0.166, ^2= 0.107,
ISO GAUSS'S METHOD OF SUBSTITUTION. [Art. 177.
6. Show that the observation equations
0.707* + 2.052)' — 2.3724r — 0.221/ = — 6-S8»
0.471* + 1-347^ — i-7iS« — 0.085/ = -~ i-63»
0.260* + o.77o_j' — 0.356a + 0.483/ = 4.40^
0.092* + 0.343^ + o.235« -f- 0.469/ = 10.21,
0.414* + 1.204J' — 1.5062 — 0.205/ = — 3-99,
0.040*+ o.i5qy + 0.104a + 0.206/= 4.34»
give rise to the normal equations
0.971* + 2.8217— 3.175a — 0.104/= — 4-815,
2.821* + 8.208;' — 9.168a — 0.251/ = — 12.961,
— 3.175* — 9.i68_j'+ 11.0282 + 0.938/= 25.697,
— 0.104* — 0.2517 + 0.938a + 0.594/ = 10.218,
and to [««] = 204.313. Determine the unknown quantities
and the probable errors of an observation.
«= —
86.41, y = 25.18, « = — 3.12, / = 17.66, ^ = 1.80.
7. Account for the small values of the weights, especially of
X and y, in Ex- 6. Show directly from the value of [bb, i] that
py < .012 and pi < .0015.
8. Ten C'bservation equations gave the normal equations
2.02530* + 0.638097 — 3.99285a = — 30.466,
0.63809* + 0.216497 — 1.12089a = — 11.959,
— 3.99285* — 1.12089^ + lo.oooooa = — 6.000,
together with [««] = 24928.; find the values and weights of the
unknown quantities and the probable errors.
§IX.] EXAMPLES. 151
X = - 202.8, y = 286.3, 2 = — 49-5;
A= -oaM. A = -0066, ^= .9119;
»" = 37-702. /-„ = 213, r, = 463, r, = 39.
9. Given the following observation equations of equal weight:
.986;« + .056)' = .000, •953'*^ + • 182;; = I 060,
.973je + .ioa)' = .S30, .943* + .219J = — .380.
^68* + .123^ = .680, .<)\()x -\- .ioiy= .200,
•959* + •157.)' = -200. .9i6* + .3i7j('= — .530,
.912* 4" 33rf = .000,
And the normal equations and the value of [««] by the method
of Art. 127. (Notice that when we put a + ^ + « + j = oas
in Art. 169 a considerable saving of labor results from the
fact that ^{fl + bY - 2{n + s)\ etc.)
8.0884*+ 1.6798;' = 1. 7160,
1.6798J1; + 0.438^)- = 0.1725,
[««] = 2.iJ22.
10. Solve the normal equations found in Ex. 9.
X = 0.642, y= — 2.07, r^ = 0.25, ry = 1.09.
11. Thirteen observation equations give the normal equa*
tions
17.50*— 6.50J'— 6.502;= 2.14,
— 6.50*+ 17.50^ — 6.502;= 13.96,
— 6.50* — 6.5oy + 20.502; = — 5.40,
[ftn] = 100.34;
find the values and probable errors of the unknown quantities
*= 0.67 ±0.60, j'= 1.17 ± 0.60, 2; = 0.32 ± 0.55-
152 GAUSS'S METHOD OF SUBSTITUTION. [An. 177.
12. Solve the normal equations
459^ — soSjc -3890 + 244^ = 507,
— 308a: + 464JI' -|-4o8a — 269/ = — 695,
— 389a: + 408,^+676^- 331/= — 653,
244^; — 269JI; —3310 + 469/ = 283,
\nn\ = 1 1 29.
X = —0.212, J = —1.471, z = —0.195, '^— — 6.4.8S;
[»z'] = io;/;,= 207, />y=i86, p^ = 2<,o, pt = 2%i.
Constants.
p = 0.4769352, log p = 9.6784603 ;
Pi/2 - 0.6744897, log P4/2 = 9.8289753 ;
Pi/TT - 0.8453475^ log Pi/7r = 8.9270353 ;
r = pV2 . e = pyn . r/.
Note that p|/2 = « + /? + ;/ + * + ..., where a = fi
VALUES OF THE PROBABILITY INTEGRAL,
OR PROBABILITY OF AN ERROR NUMERICALLY LESS THAN X,
Table I. — Values of Pi.
t=hx', Pt = -4-
t
I
2
3
4
S
6
7
8
9
o.o
0.1
0.2
0.0000
1125
2227
0.0113
1236
233s
0.0226
1348
2443
0.0338
1459
255°
0.0451
1569
2657
0.0564
1680
2763
0.0G76
1790
2869
0.0789
1900
2974
0.0901
2009
3079
0. 1 1 3
2118
3183
°-3
0.4
0.5
0.3286
4284
5205
0.3389
4380
5292
0.3491
4475
5379
0.3593
4569
5465
0.3694
4662
5549
0.3794
4755
5633
0.3893
4847
5716
0.3992
4937
5798
0.4090
5027
5879
0.4187
5117
5959
0.6
0.7
0.8
0.6039
6778
7421
0.6117
6847
7480
0.6194
6914
7538
0.6270
698.
7595
0.6346
7047
7651
0.6420
71 12
7707
0.6494
7175
7761
0.6566
7238
7814
0.6638
7300
7867
0.6708
7361
7918
0.9
1.0
I.I
0.7969
8427
8802
0.8019
8468
883s
0.8068
8508
8868
0.81 16
8548
8900
0.8163
8586
893'
0.8209
8624
8961
0.8254
8661
8991
0.8299
8698
9020
0.8342
8733
9048
0.8385
8768
9076
1.2
'•3
1.4
0.9103
9340
9523
0.9130
9361
9539
0.9155
9381
9554
0.9181
9400
9569
0.9205
9419
9583
0.9229
9438
9597
0.9252
9456
961 1
0.9275
9473
9624
0.9297
9490
9637
0.9319
9507
9649
i.S
1.6
1-7
0.9661
9763
9838
0.9673
9772
9844
0.9684
9780
985°
0.9695
9788
9856
0.9706
9796
9861
0.9716
9804
9867
0.9726
9S11
9872
0.9736
9818
9877
0.9745
9882
0.9755
9886
1.8
1.9
i.O
0.9891
9928
9953
0.9895
9931
99S5
0.9899
9934
9957
0.9903
9937
9959
0.9907
9939
9961
0.991 1
9942
9963
0.9915
9944
9964
0.9918
9947
9966
0.9922
9949
9907
0.9925
995'
9969
2.1
2.2
2-3
0.9970
9981
9989
0.9972
9982
9989
0.9973
9983
9990
0.9974
9984
9990
0.9975
9985
9991
0.9976
9985
999'
0.9977
9986
9992
0.9979
9987
9992
0.9980
9987
9992
0.9980
998S
9993
2.4
0.9993
9996
9998
0.9993
9096
999S
0.9994
9996
9998
0.9994
9997
9998
0.9994
9997
9998
0.9995
9997
9998
0.9995
9997
9998
0.9995
9997
9998
0.9995
9997
9998
0.9996
9998
9999
2.7
2.8
...
••.•
*■*
>•■
■ •■
0.9999
1 .0000
...
;:
Table II.— Values of Pt.
r p' ^ ijn '^ V^Jo
o.o
O.I
0.2
0-3
0.4
o-S
0.6
0.7
0.8
0.9
I.O
I.I
1.2
"•3
1.4
>S
1.6
1-7
1.8
1.9
2.0
2.1
2.2
2-3
2.4
2-5
2.6
2.7
2.8
2.9
3-0
31
3-2
3-3
3-4
3-
4-
I:!
0.0000 o
0538
1073
0.1604
2127
2641
o-3'43
3632
4105
0.4562
5000
5419
0.5817
6194
6550
0.6883
7195
7485
0-7753
8000
8227
0.8433
8622
8792
0.8945
9082
9205
0.9314
941 1
9495
0.9570
9635
9691
0.9740
9782
9570
0.9930
9993
9999
,0054
059
1126
0.1656
2179
2691
3.3192
3680
4x52
3.4606
S°43
5460
3.5856
6231
6584
).69is
7225
7512
'•7778
8023
8248
1.8453
8639
8808
'•8959
9095
9217
1.9324
9419
9503
'•9577
9641
9696
•9744
9786
9635
0.0108
0645
1 180
0.1709
2230
2742
0.3242
3728
4198
0.4651
5085
5500
0.5894
6267
6618
0.6947
7255
7540
0.7804
8047
8270
0.8473
8657
8824
0.8974
9108
9228
'•9943
9994
.0000
0-9334
9428
95"
0.9583
9647
9701
0.9749
9789
9691
0.9954
9995
0.0161
0699
1233
0.1761
2282
2793
0.3291
3775
4244
0.469s
5128
5540
0-5932
6303
6652
0.6979
7284
7567
0.7829
8070
8291
0.8492
8674
8839
0.0215
0752
1286
0,1814
2334
2843
0.3340
3823
4290
9121
9239
0.9344
9437
95«9
0.9590
9652
9706
0-9753
9793
9740
0.9963
9996
0.4739
5170
5581
0.5971
6339
6686
0,7011
73'3
7594
0.7854
8093
8312
0.8511
8692
8855
0.9002
9133
9250
0-9354
9446
9526
0.9597
9658
9711
0-9757
9797
9782
0.9970
9997
0.0269
0806
1339
0.1866
2385
2893
0.3389
3871
4336
0.4783
5212
5621
0.6008
6375
6719
0.7042
7343
7621
0.7879
8116
8332
0.8530
8709
8870
0.9016
9146
9261
0.9364
9454
9534
0.0323
0859
>39
0.1919
2436
2944
0.3438
39'8
4381
0.4827
5254
5660
0.6046
6410
6753
0.7073
7371
7648
0.7904
8138
8353
0.8549
8726
8886
0.9029
9158
9272
0,9603
9664
9716
0.9762
9800
9818
0.9976
9998
0.9373
9463
9541
0.9610
9669
9721
0.9766
9804
9848
0.9981
9998
0.0377
0913
1445
0.1971
2488
2994
0.3487
3965
4427
0.4871
5295
5700
0.6083
6445
6786
0.7104
7400
767s
0.7928
8161
8373
8
0.0430
0966
1498
0.2023
2539
3044
0-3535
4012
4472
0.4914
5337
5739
0.6121
6480
6818
0.7134
7428
7701
0.7952
8183
8394
0.8567
8743
8901
0.9043
9170
9283
0-9383
947"
9548
0.9616
9675
9726
0.9770
9807
9874
0.9985
9999
0.8585
8759
8916
0.9056
9182
9293
0.9392
9479
9556
0.9622
9680
973'
0.9774
981 1
9896
0.9988
9999
0.0484
1020
«55i
0.2075
2590
3093
0.3584
4059
45'7
0.4957
5378
5778
6.6157
6515
6851
0.7165
7457
7727
0.7976
8205
8414
0.8604
87:6
8930
0.9069
9194
9304
0.9401
9487
9563
0.9629
9686
9735
0.9778
9814
99'S
0.9991
9999
Number
Square.
CuDe.
Square Root.
Cube Root.
I
I
I
I.OOOO
I.OOOO
2
4
8
1.4142
1.2599
3
9
27
I.732I
1.4422
4
16
64
2.0000
1.5874
5
25
125
2.2361
1. 7100
6
36
216
2.449s
1.8171
7
49
343
2.6458
I.9129
8
64
512
2.8284
2.0000
9
81
729
3.0000
2.0801
10
I 00
I 000
3.1623
2.1544
II
I 21
» 331
3.3166
2.2240
12
I 44
I 728
3.4641
2.2894
»3
I 69
2 197
3.6056
2.35'3
H
I 90
2 744
3-7417
2.4101
IS
2 25
3 37S
3-8730
2.4662
i6
2 56
4 096
4.0000
2.5198
'7
2 89
4 9«3
4.I23I
2.S7«3
i8
3 24
5 832
4.2426
2.6207
'9
361
6859
4.3589
2.6684
20
4 00
8 000
4-4721
2.7144
21
4 4>
9 261
4.5826
2.7589
22
484
10 648
4.6904
2.8020
23
5 29
12 167
4-7958
2.8439
«4
5 76
6 25
13 824
4.8990
2.8845
*S
IS 625
5.0000
2.9240
26
6 76
17 576
5.0990
2.9625
27
7 29
19683
5.1962
3.0000
28
784
21 952
5.2915
3.0366
29
841
24 389
5.3852
3.0723
30
9 00
27 000
S-4772
3.1072
3«
9 61
29 791
5.5678
3.14H
3*
10 24
32768
5.6569
3- 1748
33
10 89
35 937
5.7446
3-2075
34
II 56
39 304
5.8310
3.2396
35
12 25
42 87s
5.9161
32711
36
12 96
46 656
6.0000
3.3019
37
13 69
SO 653
6.0828
3-3322
38
14 44
54 87a
6.1644
3-3620
39
IS 21
59 3'9
6.2450
3-3912
40
16 00
64 000
6.3246
3.4200
41
16 81
68 921
6.4031
3-4482
42
17 64
74088
6.4807
3.4760
43
18 49
79 507
6.5574
35034
44
19 36
85 184
6.6332
3-5303
45
20 25
91 125
6.7082
3.5569
46
21 16
97 336
6.7823
3-5830
47
22 09
103 823
6.8557
3.6088
48
23 04
no 592
6.9282
3.6342
49
24 01
117 649
7.0000
3.6593
SO
25 00
125 000
7.0711
3.6840
Number
Square.
Cube.
Square Root.
Cube Root.
5'
z6 01
132 651
7.14I4
3.7084
52
27 04
140 608
7.ZII1
3-7325
53
28 09
148 877
7.2801
3-7563
54
29 It)
157 464
7-3485
37798
55
30 25
166 375
7.4162
3. 80 JO
56
31 36
175 616
7-4833
3-8259
57
32 49
185 193 ^
7.5498
3.8485
58
33 64
195 112
7.6158
3.8709
59
34 81
20s 379
7.68 1 1
3893°
60
36 00
216 000
7.7460
3-9149
61
37 2i
226 981
7.8102
39365
62
38 44
238 328
7.8740
39579
63
39 69
. 250 047
7-9373
3-9791
64
40 96
262 144
8.0000
4.0000
65
42 25
274 625
8.0623
4.0207
66
43 56
287 496
8.1240
4.0412
67
44 89
300 763
8.1854
4.0615
68
46 24
314 432
8.2462
4.0817
69
47 61
328 509
8.3066
4.1016
70
49 00
343 000
8.3666
4.12.3
71
50 41
357 9"
8.4261
4.1408
72
51 84
373 248
8.4853
4.1602
73
53 29
389 017
8.5440
4-1793
74
54 76
405 224
8. 602 -J
4-1983
75
56 25
421 875
8.6603
4.2172
76
57 76
438 976
8.7178
4.2358
77
59 29
456 533
8.7750
4-2543
78
60 84
474 552
8.8318
4.2727
79
62 41
493 °39
8.8882
4.2908
80
64 00
512 000
8.9443
4.3089
81
65 61
53' 441
9.0000
4-3267
82
67 24
551 368
9-°554
4-3445
83
68 89
571 787
9.1104
4.3621
84
70 56
592 704
9.1652
4-3795
85
72 25
614 125.
9.2195
4.3968
86
73 96
636 056
9.2736
4.4140
87
75 69
658 503
9-3274
4.4310
88
77 44
681 472
9.3808
4.4480
89
79 21
704 969
94340
4.464.7
go
81 00
729 000
9.4868
4.4814
9'
82 81
753 571
9-5394
4.4979
92
84 64
778 688
9-5917
4-5144
93
86 49
804 357
9-6437
4.5307
94
88 36
830 584
9.6954
4.5468
95
90 25
857 375
9.7468
4-5629
96
92 16
884 736
9.79S0
4.5789
97
94 09
912 673
9.8489
4-5947
98
96 04
941 192
9.8995
4.6104
99
98 01
970 299
9-9499
4.6261
100
I 00 00
I 000 000
10.0000
4.6416
Number
Square.
Cube.
Square Root.
Cube Root.
lOI
I 02 01
I 030 301
10.0499
4-6570
102
I 04 04
1 061 208
10.0995
4.6723
103
I 06 09
1 092 727
10.1489
4.687 s
104
I 08 16
I 124 864
10.1980
4.7027
los
I 10 25
1 157 625
10.2470
4.7177
106
I 12 36
I 191 016
10.2956
4-73=6
107
I 14 49
I 225 043
10.3441
4-7475
108
I 16 64
I 2S9 712
10.3923
4.7622
109
I 18 81
I 29s 029
10.4403
47769
110
I 21 00
I 331 000
10.4881
4.7914
III
I 23 21
I 367 631
'0-5357
4.8059
112
I 25 44
I 404 928
10.5830
4.8203
"3
I 27 69
I 442 897
10.6301
4.8346
114
I 29 96
I 481 544
10.6771
4.8488
115
1 32 25
I 520 875
10.7238
4.8629
116
I 34 56
I 560 896
10.7703
4.8770
117
I 36 89
1 601 613
10.8167
4.8910
118
I 39 24
I 643 032
10.8628
4.9049
119
I 41 61
I 685 159
10.9087
4.9187
120
I 44 00
I 728 000
10.9545
4-9324
121
I 46 41
I 771 561
1 1.0000
4.9461
122
I 48 84
I 815 848
11.0454
4-9597
123
I SI 29
I 860 867
11.0905
4-9732
124
I S3 76
I 906 624
i'-'355
4.9866
125
' 56 25
I 953 125
11.1803
5.0000
126
I s8 76
2 000 376
11.2250
5-°i33
127
I 61 29
2 048 383
11.2694
5.026s
128
I 63 84
2 097 152
"■3137
5-°397
129
I 66 41
2 146 689
11-3578
5.0528
130
I 69 00
2 197 000
11.4018
5.0658
131
I 71 61
2 248 091
11.4455
5.0788
132
I 74 24
2 299 968
11.4891
5.0916
133
I 76 89
2 352 637
11.5326
5-1045
134
I 79 56
2 406 104
11-5758
S.1172
13s
I 82 2S
2 460 375
11.6190
5.1299
136
1 84 96
2 515 456
11.6619
5.1426
137
I 87 69
2 571 353
11.7047
5-1551
138
I 90 44
2 628 072
11-7473
S.1676
139
I 93 21
2 685 619
11.7898
5.1801
140
I 96 00
2 744 000
11.8322
5.1925
141
I 98 81
2 803 221
11.8743
5.2048
142
2 01 64
2 863 288
1 1. 9164
5.2171
143
2 04 49
2 924 207
11.9583
S-2293
144
2 07 36
2 985 984
12.0000
5-2415
145
2 10 25
3 048 625
12.0416
5- 2536
146
2 13 16
3 112 136
12.0830
5.2656
147
2 16 09
3 176 523
12.1244
5.2776
148
2 19 04
3 241 792
12.165s
5.2896
149
2 22 01
3 307 949
12.2066
5-3015
ISO
2 25 00
3 375 000
12.2474
5-3133
Number
Square,
Cube.
Square Root,
Cube Root.
IS"
2 28 01
3 442 95 «
12.2882
S-325'
152
2 31 04
3 511 808
12.3288
S-3368
153
2 34 09
3 581 577
12.3693
S-3485
154
2 37 16
3 652 264
12.4097
5-3601
'55
2 40 2S
3 723 875
12.4499
5-3717
.56
2 43 36
3 796 4«6
12.4900
5-3832
«S7
2 46 49
3 869 893
12.5300
S-3947
158
2 49 64
3 944 312
12.5698
5.4061
IS9
2 52 81
4 019 679
12.6095
S-4I7S
160
2 56 00
4 096 000
12.6491
5.4288
161
2 59 21
4 J73 281
12.6886
5.4401
i6z
2 62 44
4 251 528
12.7279
S-4SU
163
2 65 69
4 330 747
12.7671
5.4626
164
2 68 96
4 410 944
12.8062
5-4737
1 65
2 72 25
4 492 125
12.8452
5.4848
166
2 75 56
4 574 296
12.8841
5-4959
167
2 78 89
4 657 463
12.9228
5-5069
168
2 82 24
4 741 632
12.9615
5.5178
169
2 85 61
4 826 809
13.0000
5.5288
170
2 89 00
4 913 000
13.0384
5-5397
171
2 92 41
5 000 211
13.0767
5-5505
172
2 95 84
5 088 448
13.1149
S-5613
173
2 99 29
5 177 717
13.1529
5.5721
174
3 02 76
5 268 024
13.1909
5.582S
I7S
3 06 25
5 359 375
13.2288
5-5934
170
3 09 76
5 45' 776
13.2665
5.6041
177
3 >3 29
5 545 233
13-3041
5-6147
178
3 1684
5 639 752
13-3417
5.6252
179
3 20 41
5 735 339
13-3791
5-6357
180
3 24 00
5 832 000
13.4164
5.6462
181
3 27 61
5 929 741
'3-4536
5.6567
182
3 3« 24
6 028 568
13.4907
5.6671
'!3
3 34 89
6 128 487
13-5277
56774
184
3 38 56
6 229 504
13-5647
5-6877
185
3 42 25
6 331 625
13.6015
5.6980
186
3 45 96
6 434 856
13.6382
S-7083
187
3 49 69
6 539 203
136748
5.7185
188
3 53 44
6 644 672
13-7113
5.7287
189
3 57 21
6 751 269
13-7477
5-7388
190
3 61 00
6 859 000
13.7840
5-7489
191
3 64 81
6 967 871
13.8203
5-7590
192
3 68 64
7 077 888
13.8564
5-7690
193
3 72 49
7 189 057
138924
5.7790
194
3 76 36
7 301 384
13.9284
5-7890
19s
3 80 25
7 414 875
13.9642
5-7989
196
3 84 16
7 529 536
14.0000
5.8088
197
3 88 09
7 645 373
14.0357
5.8186
198
3 92 04
7 762 392
14.0712
5.8285
199
3 96 01
7 880 599
14.1067
5-8383
200
4 00 00
8 000 000
14.1421
5.8480
Number
Square.
Cube.
Square Root.
Cube Root.
201
4 04 01
8 120 601
14.1774
5.8578
202
4 08 04
8 242 408
14.2127
5.8675
203
4 12 09
8 365 427
14.2478
5.8771
204
4 16 16
8 489 664
14.2829
5.8868
20S
4 20 25
8 615 125
14.3178
5.8964
206
4 24 36
8 741 816
■4-3527
5.9059
207
4 28 49
8 869 743
'4.3875
59'5S
208
4 32 64
8 998 912
14.4222
5-9250
209
4 36 81
9 "29 329
14.4568
5.9345
210
4 41 00
9 261 000
14.4914
5-94 59
211
4 45 21
9 393 93'
14.5258
5.9533
212
4 49 44
9 528 128
14.5602
5.9627
213
4 S3 69
9 663 597
14.5945
5.9721
214
4 57 96
9 800 344
14.6287
5.9814
215
4 62 25
9 938 37 S
14.6629
5.9907
216
4 66 56
10 077 696
14.6969
6.0000
217
4 70 89
10 218 313
14.7309
6.0092
218
4 75 24
10 360 232
14.7648
6.0185
219
4 79 61
10 503 459
14.7986
6.0277
220
4 84 00
10 648 000
14.8324
6.0368
221
4 88 41
10 793 861
14.8661
6.0459
222
4 92 84
10 941 048
14.8997
6.0550
223
4 97 29
II 089 567
14-9332
6.0641
224
S 01 76
II 239 424
14.9666
6.0732
225
5 c6 25
II 390 625
15.0000
6.0822
226
S 10 76
II 543 176
15-0333
6.0912
227
S »S 29
II 697 083
15.0665
6 1002
228
5 "9 84
II 852 352
15.0997
6.091
229
5 24 41
12 008 989
15.1327
6. 1 180
230
S 29 00
12 167 000
15.1658
6.1269
231
5 33 6«
12 326 391
15.1987
6.1358
232
5 38 24
12 487 168
15-2315
6.1446
233
54289
12 649 337
15.2643
61534
234
S 47 56
12 812 904
15-297'
6.1622
23s
5 52 25
12 977 87s
'5- 3297
6 1710
236
5 56 96
13 144 256
15.3623
6.1797
237
5 61 69
13 312 053
15.3948
6.1885
238
5 66 44
13 481 272
15.4272
6.1972
239
5 7' 21
13 651 919
15.4596
6.2058
240
5 76 00
13 824 000
1 54919
6.214S
241
5 80 81
13 997 521
'5-5242
6.2231
242
5 8564
14 172 488
'5-51^3
6.2317
243
5 90 49
14 348 907
15.5885
•6.2403
244
5 95 36
14 526 784
15.6205
6.2488
24S
6 00 25
14 706 125
15.6525
6.2573
246
6 05 16
14 886 936
15.6844
6.2658
247
6 10 09
IS 069 223
15.7162
6.2743
248
6 15 04
IS 252 992
15.7480
6.2828
249
6 20 01
15 438 249
15-7797
6.2912
250
6 25 00
IS 625 000
15.8114
6.2996
Number
Square.
Cube.
Square Root.
Cube Root.
251
6 30 01
IS 813 251
15.8430
6.3080 .
252
.6 35 04
16 003 008
15.8745
6.3164
253
6 40 09
16 194 277
1 5.9060
6.3247
254
6 45 16
16 387 064
15-9374
6-3330
255
6 50 25
16 581 375
15.9687
6.3413
256
6 55 36
16 777 216
16.0000
6.3496
257
6 60 49
16 974 593
16.0312
6.3579
258
6 65 64
17 173 512
16.0624
6.3661
259
6 70 81
17 373 979
16.093s
6-3743
260
6 76 00
17 576 000
16.1245
6.3825
2O1
6 81 21
17 779 581
16.1555
6.3907
262
6 86 44
17 984 728
16.1864
6.39S8
f 263
6 91 69
18 191 447
16.2173
6.4070
264
6 96 96
18 399 744
16.2481
6.4151
26s
7 02 25
18 609 625
16.2788
6.4232
266
7 07 56
18 821 096
16.3095
6.4312
267
7 12 89
19 034 163
16.3401
6-4393
268
7 18 24
I 9 248 832
16.3707
6.4473
269
7 23 61
19 465 109
16.4012
6-4553
270
7 29 00
19 683 000
16.4317
6.4633
271
7 34 41
19 902 511
16.4621
6.4713
272
7 39 84
20 123 648
16.4924
6.4792
273
7 45 29
20 346 417
16.5227
6.4872
274
7 50 76
20 570 824
16 5529
6.4951
275
7 56 25
20 796 875
16.5831
6.5030
276
7 61 76
21 024 576
16.6132
6.5108
277
7 67 29
21 253 933
16.6433
6.5187
278
7 72 84
21 484 952
16.6733
6.5265
279
7 78 41
21 717 639
16.7033
6.5343
280
7 84 00
21 952 000
16.7332
6.5421
281
7 89 61
22 188 041
16.7631
6.5499
282
7 95 24
22 425 768
16.7929
6.5577
283
8 00 89
22 665 187
16.8226
6.5654
284
8 06 56
22 906 304
16.8523
6.5731
28s
8 12 25
23 149 125
16.8819
6.5808
286
8 17 96
23 393 656
16.9115
6.5885
287
8 23 69
23 639 903
16.9411
6.5962
288
8 29 44
23 887 872
16.9706
6.6039
289
8 35 21
24 137 569
1 7.0000
6.6115
290
8 41 00
24 389 000
17.0294
6.6191
291
8 46 81
24 642 171
17.0587
6.6267
292
8 52 64
24 897 088
17.0880
6.6343
293
. 8 58 49
25 '53 757
17.1172
6.6419
294
8 64 36
25 412 184
17.1464
6.6494
295
8 70 25
25 672 375
17.1756
6.6569
296
8 76 16
25 934 336
17.2047
6.6644
297
8 82 09
26 198 073
17-2337
6.6719 ,
298
8 88 04
26 463 592
17.2627
6.6794
299
8 94 01
26 730 899
17.2916
6.6869 ,
300
9 00 00
27 000 000
17.3205
6.6943
Number
Square.
Cube.
Square Root.
Cube Root.
301
302
303
304
30s
9 06 01
9 12 04
9 18 09
9 24 16
9 30 25
27 270,901
27 543 608
27 818 127
28 094 464
28 372 625
17-3494
17-3781
17.4069
17-4356
17.4642
6.7018
6.7092
6.7166
6.7240
6.73'3
306
3°7
308
309
310
9 36 36
9 42 49
9 48 64
9 54 81
9 61 00
28 652 616
28 934 443
29 218 112
29 503 629
29 791 000
17.4929
17-5214
17-5499
17.5784
17.6068
6.7387
6.7460
6-7533
6.7606
6.7679
3"
312
313
3«4
315
9 67 21
9 73 44
9 79 69
9 85 96
9 92 25
30 080 231
30 371 328
30 664 297
30 959 144
31 255 875
17.6352
17.6635
17.6918
17.7200
17.7482
(=.7752
0.7824
6.7897
6.7969
6.8041
316
317
318
319
320
9 98 56
10 04 89
10 II 24
10 17 61
10 24 00
31 554 496
31 85s 013
32 157 432
32 461 759
32 768 000
17.7764
17.8045
17.8326
17.8606
17.8885
6.8113
6.8185
6.8256
6.8328
6.8399
321
322
323
324
325
10 30 41
10 36 84
10 43 29
10 49 76
10 56 25
33 076 161
33 386 248
33 698 267
34 012 224
34 328 125
17.9165
17.9444
17.9722
18.0000
18.0278
6.8470
6.8541
6.8612
6.8683
6.8753
326
327
328
329
330
10 62 76
10 69 29
10 75 84
,10 82 41
'10 89 00
34 645 976
34 965 783
35 287 552
35 611 289
35 937 000
18-0555
18.0831
18.II08
18.1384
18.1659
6.8824
6.8894
6.8964
6.9034
6.9104
33'
332
333
334
33S
10 95 61
11 02 24
II 08 89
II IS 56
II 22 25
36 264 691
36 594 368
36 926 037
37 259 704
37 595 375
18.1934
18.2209
18.2483
18.3757
18.3030
6-9174
6.9244
6.9313
6.9382
6-945'
336
337
338
339
340
II 28 96
II 35 69
II 42 44
II 49 21
II 56 00
37 933 056
38 272 753
38 614 472
38 958 219
39 304 000
i8-33°3
18.3576
18.3848
18.4120
18.439'
6.9521
6.9589
6.9658
6.9727
6-9795
341
342
343
344
345
II 62 81
II 69 64
II 76 49
II 83 36
n 90 25
39 651 821
40 001 688
40 353 607
40 707 584
41 063 625
18.4662
18.4932
18.5203
18.5472
18.5742
6.9864
6.9932
7.0000
7.0068
7.0136
346
347
348
349
35°
11 97 16
12 04 09
12 II 04
12 18 01
12 25 00
41 421 736
41 781 923
42 144 19a
42 508 549
42 87s 000
18.601 1
18.6279
18.6548
18.6815
18.7083
7.0203
7.0271
7-0338
7.0406
7-0473
Number
Square,
Cube.
Square Root.
Cube Root.
351
352
353
354
355
12 32 01
12 39 04
12 46 09
12 53 16
12 60 25
43 243 55 «
43 614 208
43 986 977
44 361 864
44 738 87s
18.7350
18.7617
18.7883
18.8149
18.8414
7.0540
7.0607
7.0674
7.0740
7.0807
3S6
357
3S8
359
300
12 67 36
12 74 49
12 81 64
12 88 8i
12 96 00
4S 118 016
45 499 293
45 882 712
46 268 279
46 656 000
18.8680
18.8944
18.9209
18.9473
18.9737
7-0873
7.0940
7.1006
7.1072
7.1138
361
362
363
364
365
13 03 21
13 10 44
13 17 69
13 24 96
13 32 25
47 04S 881
47 437 928
47 832 147
48 228 S44
48 627 I2S
19.0000
19.0263
19.0526
19.0788
19.1050
7.1204
7.1269
7-I33S
7.1400
7.1466
366
367
368
369
370
13 39 56
13 46 89
13 54 24
13 61 61
13 69 00
49 027 896
49 430 863
49 836 032
50 243 409
SO 653 000
19-13"
19.1572
19-1833
19.2094
19.2354
7-1531
7.1596
7.1661
7-1726
7.1791
371
372
373
374
375
13 76 41
13 83 84
13 91 29
13 98 76
14 06 25
SI 064 811
SI 478 848
51 89s 117
52 313 624
52 734 375
19.2614
19.2873
'9-3'32
>9-339i
19.3649
7. "855
7.1920
7.1984
7.2048
7.2112
376
377
378
379
380
H 13 76
14 21 29
14 28 $4
14 36 41
14 44 00
53 >57 376
53 582 633
54 010 IS2
54 439 939
S4 872 000
19.3907
19.4165
19.4422
19.4679
19.4936
7.2177
7.2240
7.2304
7,2368
7.2432
3f'
382
383
384
385
14 51 61
14 59 24
14 66 89
14 74 56 -
14 82 25
55 306 341
ss 742 968
56 181 887
56 623 104
57 066 625
19.5192
19.5448
19.5704
19-5959
19.6214
7-2495
7-2558
7.2622
7.268s
7-2748
386
387
388
389
390
14 89 96
14 97 69
15 05 44
IS 13 21
IS 21 00
57 5'2 456
57 960 603
58 411 072
58 863 869
59 319 000
19.6469
19.6723
19.6977
19.7231
19.7484
7.281 1
7.2874
7.2936
7-2999
7.3061
39'
392
393
394
395
IS 28 81
IS 36 64
IS 44 49
IS 52 36
15 60 25
59 776 471
60 236 288
60 698 457
61 162 984
61 629 §75
19-7737
19.7990
19.8242
19-8494
19.8746
7-3'24
7-3'86
7.3248
7-33>o
7-3372
396
397
398
399
400
IS 68 16
15 76 09
IS 84 04
15 92 01
16 00 00
62 099 136
62 570 773
63 044 792
63 521 199
64 000 000
19.8997
19.9249
19.9499
19-9750
20.0000
7-3434
7-3496
7-3558
7.3619
7.3681
Number
Square.
Cube.
Square Root. Cube Root. |
401
402
H03
404
40s
16 08 01
16 16 04
16 24 09
16 32 16
16 40 25
64 481 201
64 964 808
65 450 827
65 939 264
66 430 125
20.0250
20.0499
20.0749
20.0998
20.1246
7-3742
7-3803
7.3864
7-3925
7.3986
406
409
410
16 48 36
16 56 49
16 64 64
16 72 81
16 81 00
66 923 416
67 419 143
67 917 312
68 417 929
68 921 000
20.1494
20.1742
20.1990
20.2237
20.2485
7.4047
7.4108
7.4169
7.4229
7.4290
411
41 J
4«3
414
41S
16 89 21
16 97 44
17 0569
17 13 96
17 22 25
69 426 531
69 934 528
70 444 997
70 9S7 944
71 473 375
20.2731
20.2978
20.3224
20.3470
20.3715
7-4350
7.4410
7-4470
7-4530
7-4590
416
417
418
419
420
17 30 56
17 38 89
17 47 24
17 SS 61
17 64 00
71 991 296
72 511 713
73 034 632
73 560 059
74 088 000
20.3961
20.4206
20.4450
20.469s
20.4939
7.4650
7.4710
7-4770
7.4829
7.4889
421
422
424
42s
17 72 41
17 80 84
17 89 29
17 97 76
18 06 25
74 618 461
75 IS" 448
75 686 967
76 225 024
76 765 625
20.5183
20.5426
20.5670
20.5913
20.6155
7.4948
7:5007
7.5067
7.5126
7-5'85
426
427
428
429
430
18 14 76
18 23 29
18 31 84
18 40 41
18 49 00
77 308 776
77 854 483
78 402 752
78 953 589
79 507 000
20.6398
20.6640
20.6882
20.7123
20.7364
7-5244
7.5302
7-5361
7-5420
7-5478
43'
432
433
434
43S
18 S7 61
18 66 24
18 74 89
18 83 56
18 92 25
80 062 991
80 621 568
81 182 737
81 746 504
82 312 875
20.7605
20.7846
20.8087
20.8327
20.8567
7-5537
7-SS9S
7-5654
7.5712
7-5770
436
437
438
439
440
19 00 96
19 09 69
19 18 44
19 27 21
19 36 00
82 881 856
83 453 453
84 027 672
84 604 519
85 184 000
20.8806
20.9045
20.9284
20.9523
XO.9762
7.5828
7.5886
7-5944
7.6001
7.6059
441
442
443
444
445
19 44 81
•9 53 64
19 62 49
19 71 36
19 80 25
85 766 121
86 350 888
86 938 307
87 528 384
88 121 125
21,0000
21.0238
21.0476
21.0713
21.0950
7.61 17
7.6174
7.6232
7.6289
7-6346
446
447
448
449
450
19 89 16
199809
20 07 04
20 16 01
20 25 00
88 716 536
89 314 623
89 915 392
90 518 849
91 125 000
21.1187
21.1424
21. 1660
21.1896
21.2132
7.6403
7.6460
7.6517
7.6574
7-6631
Number
Square.
Cube.
Square Root.
Cube Root.
4S2
453
454
455
20 34 01
20 43 04
20 52 09
ZO 01 16
20 70 25
9' 733 851
92 345 408
92 959 677
93 576 664
94 196 375
21.2368
21.2603
21.2838
21.3073
21.3307
7.668S
7.6744
7.6801
7.6857
7.6914
456
457
458
459
460
20 79 36
20 88 49
20 97 64
21 06 81
21 16 00
94 818 816
95 443 993
96 071 912
96 702 579
97 336 000
21-3542
21.3776
21.4009
21.4243
21.4476
7.6970
7.7026
7.7082
7-7138
7.7194 :
461
462
463
464
465
21 25 21
21 34 44
21 43 69
21 52 96
21 62 25
97 972 181
98 611 128
99 252 847
99 897 344
100 544 625
21.4709
21.4942
21.5174
21.5407
21.5639
7.7250
7-7306
7.7362 !
7-7418 i
7-7473 ;
466
467
468
469
470
21 71 56
21 80 89
21 90 24
21 99 61
22 09 00
101 194 696
loi 847 563
102 503 232
103 161 709
103 823 000
' 21.5870
21.6102
21.6333
21.6564
21.6795
7.7529 ;
7-7584 :
7.7639
7.7695
7.7750
471
472
473
474
47S
22 18 41
22 27 84
22 37 29
22 46 76
22 56 25
104 487 III
105 154 048
105 823 817
106 496 424
107 171 875
21.7025
21.7256
21.7486
21.7715
21-7945
7.7805
7.7860
7-7915
7.7970
7.8025
476
477
478
479
480
22 65 76
22 75 29
22 84 84
22 94 41
23 04 CO
107 850 176
i°8 531 333
109 215 352
109 902 239
no 592 000
21.8174
21.8403
21.8632
21.8861
21.9089
7.8079
7-8134
7.8188
7-8243
7.8297
481
482
483
484
485
23 13 61
23 23 24
23 32 89
23 42 56
23 52 25
III 284 641
111 980 16S
112 678 587
"3 379 904
114 084 125
21.9317
21-9545
21.9773
22.0000
22.0227
7-8352
7.8406
7.8460
7.8514
7.8568
486
487
488
489
490
23 61 9S
23 71 69
23 81 44
23 91 21
24 01 00
114 791 256
115 501 303
116 214 272
116 930 169
117 649 000
22.0454
22.068l
22.0907
22.1133
22.1359
7.8622
7.8676
7.8730
7-8784
7-8837
491
492
493
494
495
24 10 81
24 20 64
M 30 49
24 40 36
24 50 25
118 370 771
119 095 '488
119 823 157
J 20 553 784
121 287 375
22.1585
22.1811
22.2036
22.2261
22.2486 .
7.8891
7-8944
7.8998
7.9051
7.9105
496
497
498
499
500
24 60 16
24 70 09
24 80 04
24 90 01
25 00 00
122 023 936
J22 763 473
123 S°5 992
124 251 499
125 000 000
22.27H
22.2935
22.3159
22.3383
22.3607
7.9158
7.9211
7.9264
7-9317
7.9370
\Nuinbn
Square.
Cube.
Square Root.
Cube Root.
501
S02
503
504
505
- 25 10 01
25 20 04
25 30 09
25 40 16
25 50 25
125 751 501
■126 506 008
127 263 527
128 024 064
128 787 625
22.3830
22.4054
22.4277
22.4499
22.4722
7-9423
7.9476
7.9528
7.9581
7-9634
5o6
507
508
509
510
25 60 36
25 70 49
25 80 63
25 90 81
26 01 CO
129 554 216
130 323 843
131 096 512
131 872 229
132 651 000
22.4944
22.5167
22.5389
22.5610
22.5832
7.9686
7-9739
7.979'
7-9843
7.9896
5"
512
513
514
515
26 II 21
26 21 44
26 31 69
26 41 96
26 52 25
133 432 831
134 217 728
13s 005 697
13s 796 744
136 590 87s
22.6053
22.6274
22.6495
22.6716
22.6936
7.9948
8.0000
8.0052
8.0104
8.0156
5.6
518
519
520
26 62 56
26 72 89
26 83 24
26 93 61
27 04 00
137 388 096
138 188 413
138 991 832
139 798 359
140 608 000
22.7156
22.7376
22.7596
22.7816
22.8035
8.0208
8.0260
8.0311
8.0363
8 0415
521
522
523
524
525
27 14 41
27 24 84
27 35 29
27 45 76
27 56 25
141 420 761
142 236 648
143 055 667
143 877 824
144 703 125
22.8254
22.8473
22.8692
22.8910
22.9129
8.0466
8.0517
8.0569
8.0620
8.0671
526
527
528
529
'53°
27 66 76
27 77 29
27 87 84
27 98 41
28 09 00
M5 531 576
146 363 183
147 197 952
148 035 889
148 877 000
22.9347
22.9565
22.9783
23.0000
23.0217
8.0723
8.0774
8.0825
8.0876
8.0927
531
532
533
534
535
28 19 61
28 30 24
28 40 89
28 51 56
28 62 25
149 721 291
150 568 768
151 419 437
152 273 304
153 130 375
23-0434
23.0651
23.0868
23.1084
23.1301
8.0978
8.1028
8.1079
8.1130
8.1 180
536
537
538
539
540
28 72 96
28 83 69
28 94 44
29 05 21
29 16 00
■S3 990 656
1 54 854 153
155 720 872
156 590 819
157 464 000
23. 1 5' 7
23-1733
23.1948
23.2164
23-2379
8.1231
8.1281
8.1332
8.1382
8.r433
541
542
543
544
545
29 26 81
29 37 64
29 48 49
29 59 36
29 70 25
158 340 421
159 220 088
160 103 007
160 989 184
i6i 878 625
23-2594
23.2809
23.3024
23.3238
23-3452
8.1483
c-'533
8.1583
8-1633
8.1683
546
548
549
55°
29 81 16
29 92 09
30 03 04
30 14 01
30 25 00
162 771 336
163 667 323
164 566 592
165 469 149
166 375 000
23.3666
23.3880
23.4094
23-4307
23.4521
8.1733
8.1783
8.1833
8.1882
8.1932
Number
Square.
Cube.
Square Root.
Cube Root. '
55'
5S»
553
554
555
-50 36 01
30 47 04
30 58 09
30 69 16
30 80 25
167 284 151
168 196 608
169 112 377
170 031 464
170 953 87s
23-4734
23-4947
23.5160
23-5372
23-5584
8.1982 ,
8.2031
8.2081
8.2130
8.2180
556
558
560
30 91 36
31 02 49
31 13 64
31 24 81
31 36 00
171 879 616
172 808 693
173 741 112
174 676 879
175 616 000
23-5797
23.6008
23.6220
23.6432
23.6643
8.2229
8.2278
8.2327
8.2377
8.2426
562
563
56s
31 47 21
3' 58 44
31 69 69
31 80 96
31 92 25
176 558 481
177 504 328
•78 453 547
179 406 144
180 362 125
23.6854
23.7065
23.7276
23-7487
23.7697
8.2475
8.2524
8-2573
8.2621
8.2670
566
567
569
570
32 °3 56
32 14 89
32 26 24
32 37 61
32 49 00
181 321 496
182 284 263
183 250 432
184 220 009
185 193 000
23.7908
23.8118
23.8328
23-8537
23.8747
8.2719
8.2768
8.2816
8.2865
8.29.3
57'
572
573
574
575
32 60 41
32 71 84
32 83 29
32 94 76
33 06 25
186 169 411
187 149 248
188 132 517
189 119 224
190 109 375
23.8956
23.9165
23-9374
23-9583
23-9792
8.2962
8.3010
8.3059
8.3107
8-3>55
576
577
578
580
33 >7 76
33 29 29
33 40 84
33 52 41
33 64 00
191 102 976
192 100 033
193 100 552
194 104 539
195 112 00.0
24.0000
24.0208
24.0416
24.0624
24.0832
8.3203
8.3251
8.3300
8.3348
8-3396
5f'
583
584
585
33 75 61
33 87 24
33 98 89
34 »o 56
34 22 25
196 122 941
197 137 368
198 155 287
199 176 704
200 201 625
24.1039
24.1247
24.1454
24.1661
24.1868
8.3443
8.3491
8-3539
8.3587
8.3634
586
587
588
589
590
34 33 96
34 45 69
34 57 44
34 69 21
34 81 00
201 230 056
202 262 003
203 297 472
204 336 469
205 379 000
24.2074
24.2281
24.2487
24.2693
24.2899
8.3682
8.3730
8.3777
8.3825
8.3872
8.3919
8.3967
8.4014
8.4061
8.4108
591
592
593
594
595
34 92 81
35 04 64
35 16 49
35 28 36
35 40 25
206 425 071
207 474 688
208 527 857
209 584 584
210 644 875
24.3105
24-3311
24.3516
24-3721
24.3926
596
597
598
35 52 16
35 64 09
35 76 04
35 88 01
36 00 00
211 708 736
212 776 173
213 847 192
214 921 799
216 000 000
24.4131
24-4336
24.4540
24-4745
24-4949
8.415s
8.4202
8.4249
8.4296
8-4343
Number
Square.
Cube.
Square Root.
Cube Root.
60 1
36 12 01
217 081 801
24.5153
8.4390
602
36 24 04
218 167 208
24-5357
8.4437
603
36 36 09
219 256 227
24.5561
8.4484
604
36 48 16
220 348 864
24.5764
8.4530
60s
606
36 60 25
221 445 125
24.5967
8.4577
36 72 36
222 545 016
24.6171
8.4623
607
36 84 49
223 648 543
24.6374
8.4670
608
36 96 64
224 755 712
24.6577
8.4716
609
37 08 81
225 866 529
24.6779
8.4763
610
37 21 00
226 981 000
24.6982
8.4809
6.1
37 33 21
228 099 131
24.7184
8.4856
612
37 45 44
229 220 928
24.7386
8.4902
613
37 57 69
230 346 397
24.7588
8.4948
614
37 69 96
231 475 544
24.7790
8.4994
6.5
616
37 82 25
232 608 375
24.7992
8.5040
37 94 56
233 744 896
24.8193
8.5086
617
38 06 89
234 885 113
24.8395
8.5132
618
38 19 24
236 029 032
24.8596
8.5178
619
38 31 6t
237 176 659
24 8797
8.5224
620
621
38 44 00
238 328 000
24.8998
8.5270
38 56 41
239 483 061
24.9199
8.5316
622
38 68 84
240 641 848
24-9399
8.5362
623
38 81 29
241 804 367
24.9600
8.5408
624
38 93 76
242 970 624
24.9800
8.5453
625
39 06 25
244 140 625
25.0000
8.5499
626
39 18 76
245 314 376
25.0200
8.5544
627
39 3' 29
246 491 883
25.0400
8.5590
628
39 43 84
247 673 152
25.0599
8.5635
629
39 56 41
248 858 189
25.0799
8.5681
6^0
39 69 00
250 047 000
25.0998
8.5726
631
39 81 61
251 239 591
25.1197
8.5772
632
39 94 24
252 435 968
25.1396
8.5817
6J3
40 06 89
253 636 137
25.1595
8.5862
634
40 19 56
254 840 104
25.1794
8.5907
.63s
40 32 25
256 047 87s
25.1992
8.5952
636
40 44 96
257 259 456
25.2190
8.5997
637
40 57 69
258 474 853
25.2389
8.6043
638
40 70 44
259 694 072
25.2587
8.6088
639
40 83 21
260 917 119
25.2784
8.6132
640
40 g6 00
262 144 000
25.2982
8.6177
641
41 08 81
263 374 721
25.31S0
8.6222
642
41 21 64
264 609 288
25-3377
8.6267
643
41 34 49
265 847 707
25-3574
8.6312
644
41 47 36
267 089 984
25-3772
8.6357
64s
41 60 25
268 336 125
25.3969
8.6401
646
41 73 16
269 586 136
25.4165
86446
647
41 86 09
270 840 023
25.4362
8.6490
648
41 99 04
272 097 792
25.4558
8.6535
649
42 12 01
273 359 549
25-4755
8.6579
650
42 25 00
274 625 000
2549S1
8.6624
Number
651
652
653
654
65s
Square.
Cube.
Square Root.
Cube Root.
42 38 01
42 51 04
42 64 09
42 77 16
42 90 25
27s 894 451
277 167 808
278 445 077
279 726 264
281 oil 37S
2S-S'47
25-5343
25-5539
25-5734
25-5930
8.6668
8.6713
8.6757
8.6801
8.6845
656
657
658
43 03 36
43 '6 49
43 29 64
43 42 81
43 56 00
282 30c 416
283 593 393
284 890 312
286 191 179
287 496 000
25.6125
25.6320
25.6515
25.6710
25 690s
8.6890
8.6934
8.6978
8.7022
8.7066
661
662
663
664
665
43 69 21
43 82 44
43 95 69
44 08 96
44 22 25
288 804 781
290 117 528
291 434 247
292 754 944
294 079 625
25.7099
25.7294
25.7488
25.7682
25.7876
8.7110
8.7154
8.7198
8.7241
8.7285
666
667
668
669
670
44 3S .56
44 48 89
44 62 24
44 75 6r
44 89 00
295 408 296
296 740 963
298 077 632
299 418 309
300 763 000
25.8070
25.8263
25-8457
25.8650
25.8844
8.7329
8-7373
8.7416
8.7460
8.7503
671
672
673
674
675
45 02 41
45 15 84
45 29 29
45 42 76
45 56 25
302 III 711
303 464 448
304 821 217
306 182 024
307 546 875
25.9037
25.9230
25.9422
, 25.9615
25.9808
8.7547
8.7590
8.7634
8.7677
8.7721
676
677
678
679
680
45 69 76
45 83 29
45 96 84
46 10 41
46 24 00
308 915 776
310 288 733
311 665 752
313 046 839
314 432 000
26.0000
'26.0192
26.0384
26.0576
26.076S
8.7764
8.7807
8.7850
8.7893
8.7937
681
682
683
684
68s
46 37 6i
46 51 24
46 64 89
46 78 56
46 92 25
315 821 241
317 214 568
318 611 987
320 013 504
321 419 125
26.0960
26. II 5 1
26.1343
26.1534
26.1725
8.7980
8.802 1
8.8066
8.8109
8.8152
686
687
688
689
690
47 OS 96
47 19 69
47 33 44
47 47 21
47 61 00
322 828 856
324 242 703
325 660 672
327 082 769
328 509 000
26.1916
26.2107
26.2298
26.2488
26.2679
8.8194
8.8237
8.8280
8.8323
8.8366
691
692
693
694
695
47 74 81
, 47 88 64
48 02 49
48 16 36
48 30 25
329 939 371
331 373 888
332 812 557
334 25s 384
335 702 375
26.2869
26.3059
26.3249
26.3439
26.3629
8.8408
8.8451
8.8493
8.8536
8.8578
696
697
698
699
700
48 44 16
48 58 09
48 72 04
48 86 01
49 00 00
337 153 536
338 608 873
340 068 392
341 532 099
343 000 000
26.3818
26.4008
26.4197
26.4386
26.4575
8.8621
8.8663
8.8706
8.8748
8.8790 1
Numbei
Square,
Cube.
Square Root.
Cube Root.
701
49 14 01
344 472 lOI
26.4764
8.8833
702
49 28 04
345 948 408
26.4953
8.8875
703
49 42 09
347 428 927
26.5141
8.8917
704
49 56 16
348 913 664
26,5330
8.8959
70s
49 70 25
350 402 625
26.5518
8.9001
706
49 84 36
351 895 816
26.5707
8.9043
707
49 98 49
353 393 243
26.5895
8.9085
708
50 12 64
354 894 912
26.6083
8.9127
709
50 26 81
356 400 829
26.6271
8.9169
710
50 41 00
357 911 000
26.6458
8.9211
7"
5° 55 21
359 425 431
26.6646
8.9253
712
50 69 44
360 944 128
26.6833
8.9295
713
50 83 69
362 467 097
26,7021
8-9337
714
5° 97 96
363 994 344
26,7208
8.9378
715
51 12 25
365 525 875
26.7395
8.9420
716
51 26 56
367 061 696
26,7582
8.9462
717
51 40 89
368 601 813
26,7769
8.9503
718
■ 51 55 24
370 146 232
26,7955
8.9545
719
51 69 61
371 694 959
26,8142
8.9587
720
51 84 00
373 248 000
26,8328
8.9628
721
51 98 41
374 805 361
26.8514
8.9670
722
52 12 84
376 367 048
26.8701
8.971 1
723
52 27 29
377 933 °67
26.8887
8.9752
724
52 41 76
379 5°3 424
26.9072
8-9794
725
52 56 25
381 078 125
26.9258
8.983s
726
52 70 76
382 657 176
26.9444
8.9876
727
52 85 29
384 240 583
26.9629
8.9918
728
52 99 84
385 828 352
26.9815
8.9959
729
S3 14 4'
387 420 489
27,0000
9.0000
730
53 29 00
389 017 000
27,0185
9.0041
731
S3 43 61
390 617 891
27.0370
9.0082
732
S3 58 24
392 223 168
27-0555
9.0123
733
S3 72 89
393 832 837
27.0740
9,0164
734
53 87 56
395 446 904
27.0924
9.0205
735
54 02 25
397 065 375
27.1109
9.0246
736
54 16 96
398 688 256
27,1293
9^287
737
54 3J 69
400 315 553
27,1477
9.0328
738
54 46 44
401 947 272
27.1662
9.0369
739
54 61 21
403 583 419
27.1846
9.0410
740
54 76 00
405 224 000
27.2029
9.0450
741
54 90 81
406 869 021
27.2213
9.0491
742
55 05 64
408 518 488
27.2397
9-0532
743
55 20 49
410 172 407
27.2580
9.0572
744
55 35 36
411 830 784
27.2764
9.0613
745
55 5° 25
55 65 16
413 493 625
27.2947
9.0654
746
415 160 936
27.3130
9.0694
747
55 80 09
416 832 723
27-3313
9-°735
748
55 95 04
418 508 992
27.3496
9-0775
749
56 10 01
420 189 749
27.3679
9.0816
75°
56 25 00
421 875 000
27.3861
9.0856
Number
Square.
Cube.
Square Root.
Cube Root.
75'
752
753
754
755
56 40 01
56 55 04
56 70 09
56 85 16
57 00 25
423 564 751
425 259 008
426 957 777
428 661 064
430 368 875
27.4044
27.4226
27.4408
27.4591
27-4773
9.0896
9.0937
9.0977
9.IO17
9.1057
756
"^
758
759
760
57 T5 36
57 3° 49
57 45 64
57 60 81
57 76 00
432 081 216
433 798 093
435 S>9 512
437 245 479
438 976 000
27.4955
27.5136
27.5318
27.5500
27.5681
9.1098
9.1 138
9.I178
9.I218
9.1258
761
762
763
764
765
57 91 21
58 06 44
58 21 69
58 36 96
58 52 25
440 711 081
442 45° 728
444 194 947
445 943 744
447 697 125
27.5862
27.6043
27.6225
27.6405
27.6586
9.1298
9- '338
9.1378
9.1418
9.1458
766
767
768
769
770
58 67 56
58 82 89
58 98 24
59 '3 61
59 29 00
449 455 096
451 217 663
452 984 832
454 756 609
456 533 o°o
27.6767
27.6948
27.7128
27.7308
27.7489
9.1498
9- "537
9-»577
9.1617
9.1657
771
772
773
774
775
59 44 41
59 59 84
59 75 29
59 9° 76
60 06 25
458 314 on
460 099 648
461 889 917
463 684 824
465 484 375
27.7669
27.7849
27.8029
27.8209
27.8388
9.1696
9-1736
9-1775
9.181S
9.1855
776
777
778
779
780
60 21 76
60 37 29
60 52 84
60 68 41
60 84 00
467 288 576
469 097 433
470 910 952
472 729 139
474 552 000
27.8568
27.8747
27.8927
27.9106
27.9285
9.1894
9-1933
9-1973
9.2012
9.2052
781
782
783
784
785
60 99 61
61 15 24
61 30 89
61 46 56
61 62 25
476 379 541
478 211 768
480 048 687
481 890 304
483 736 625
27.9464
27.9643
27.9821
28.0000
28.0179
9.2091
9.2130
g.2170
9.2209
9.2248
786
787
788
789
790
61 77 96
61 93 69
62 09 44
62 25 21
62 41 00
485 587 656
487 443 403
489 303 872
491 169 069
493 039 00°
28.0357
28.0535
28.0713
28.0891
28.1069
9.2287
9.2326
9.2365
9.2404
9-2443
791
792
793
794
795
62 56 81
62 72 64
62 88 49
63 04 36
63 20 25
494 9'3 671
496 793 088
498 677 257
500 566 184
502 459 875
28.1247
28.1425
28.1603
28.1780
28.1957
9.2482
9.2521
9.2560
9-2599
9.2638
796
797
798
799
800
63 36 16
63 52 09
63 68 04
63 84 01
64 00 00
504 358 336
506 261 573
508 169 592
510 082 399
512 000 000
28.213s
28.2312
28.2489
28.2666
28.2843
9.2677
9.2716
9.2754
9-2793
9.2832
Number
Square.
Cube.
Square Root.
Cube Root
80 1
802
803
804
805
64 16 01
64 32 04
64 48 09
64 64 16
64 80 25
513 922 401
515 849 608
517 781 627
519 718 464
521 660 125
28.3019
28.3196
28.3373
28.3549
28.3725
9.2870
9.2909
9.2948
9.2986
9.3025
806
807
808
809
810
64 96 36
65 12 49
65 28 64
65 44 81
65 61 CO
523 606 616
525 SS7 943
527 514 112
529 475 129
53 ( 441 000
28.3901
28.4077
28.4253
28.4429
28.4605
9.3063
9.3102
9.3140
9-3 '79
9.3217
8[i
812
814
815
65 77 21
65 93 44
66 09 69
66 25 96
66 42 25
533 4>i 731
S3S 387 328
537 367 797
539 353 144
541 343 375
28.4781
28.4956
28.5132
28.5307
28.5482
9-3255
9-3294
9-3332
9-3370
9.3408
816
8>7
818
819
820
66 58 56
66 74 89
66 9 [ 24
67 07 61
67 24 GO
543 338 496
545 338 513
547 343 432
549 353 259
551 368 000
28.5657
28.5832
28.6007
28.6182
28.6356
9-3447
9-3485
93523
9-3561
9-3599
82[
822
823
824
825
67 40 41
67 56 84
67 73 29
67 89 76
68 06 25
553 387 661
555 4>2 248
557 441 767
559 476 224
561 51S 625
28.6531
28.6705
28.6880
28.7054
28.7228
9-3637
9-3675
93713
9-3751
9-3789
826
827
828
829
830
68 22 76
68 39 29
68 55 84
68 72 41
68 89 00
563 SS9 976
565 609 283
567 663 552
569 722 789
571 787 000
28.7402
28.7576
28.7750
28.7924
28.8097
9-3827
9-3865
9.3902
9-3940
9-3978
831
832
834
835
69 05 6i
69 22 24
69 38 89
69 55 56
69 72 25
573 856 191
575 930 368
578 009 537
580 093 704
582 182 875
28.8271
28.8444
28.8617
28.8791
28.8964
9.4016
9-4053
9.4091
9.4129
9.4166
836
837
838
839
840
69 88 96
70 05 69
70 22 44
70 39 21
70 56 00
584 277 056
586 376 253
588 480 472
590 589 719
592 704 000
28.9137
28.9310
28.9482
28.965s
28.9828
9.4204
9.4241
9.4279
9.4316
9-4354
841
842
843
844
84 s
70 72 81
70 89 64
71 06 49
71 23 36
71 40 25
594 823 321
596 947 688
599 077 107
601 211 584
603 351 125
29.0000
29.0172
29.0345
29.0517
29.0689
9-4391
9-4429
9.4466
9-4503
9-4541
846
847
848
849
850
71 57 16
71 74 09
71 91 04
72 08 01
72 25 00
60s 495 736
607 645 423
609 800 192
611 960 049
614 125 000
29.0861
29-1033
29.1204
29.1376
29.1548
9.4578
9.4615
9.4652
9.4690
9-4727
Number
Square.
Cube.
Square Root,
Cube Root.
8s.
72 42 OI
616 295 051
29.1719
9.4764
852
72 59 04
618 470 208
29.1890
9.4801
853
72 76 09
620 650 477
29.2062
9.4838
854
72 93 16
622 835 864
29.2233
9.4875
855
73 10 25
625 026 375
29.2404
9.4912
' 856
73 27 36
627 222 016
29.2575
9.4949
i 857
73 44 49
629 422 793
29.2746
94986
• 858
73 61 64
631 628 712
29.2916
9-5023
i ^P
73 78 81
633 839 779
29.3087
9.5060
i 860
_ 73 96 00
636 056 000
29. ',258
9.5097
i 86[
74 13 21
638 277 381
29.3428
9-5134
! 862
74 30 44
640 503 928
29.3598
9.517I
! IP
74 47 69
642 735 647
29.3769
9.5207
i 864
74 64 96
644 972 544
29.3939
9-5244
865
74 82 25
647 214 625
29.4109
9.5281
866
74 99 56
649 461 896
29.4279
9-S3'7
j 867
75 16 89
651 714 363
29.4449
9-5354
1 868
75 34 24
653 972 032
29.4618
9-539>
869
75 5' 61
656 234 909
29.4788
9-5427
870
75 69 00
658 503 000
29.4958
9.5464
871
75 86 41
660 776 311
29.5127
9.5501
872
76 03 84
663 054 848
29.5296
95537
873
76 21 29
665 338 617
29.5466
9-5574
874
76 38 76
667 627 624
29.5635
9.5610
875
76 56 25
669 921 875
29.5804
5.5647
876
76 73 76
672 221 376
29-5973
9.5683
877
76 91 29
674 526 133
29.6142
9-5719
878
77 08 84
676 836 152
29.6311
9-5756
879
77 26 41
679 151 439
29.6479
9.5792
880
77 44 00
68 I 472 000
29.6648
9.5828
881
77 61 61
683 797 841
29.6816
9.5865
882
77 79 24
686 128 968
29.6985
9.5901
883
77 96 89
688 465 387
29-7153
9-5937
884
78 14 56
690 807 104
29.7321
9-5973
885
78 32 25
693 154 125
29.7489
9.6010
886
78 49 96
695 506 456
29.7658
9.6046
887
78 67 69
697 864 103
29.7825
9.6082
888
78 85 44
700 227 072
29.7993
9.61 18
889
79 03 21
702 595 369
29.8161
9.6154
890
79 21 00
704 969 000
29.8329
9.6190
891
79 38 81
707 347 971
J29.8496
. 9.6226
892
79 56 64
709 732 288
29.8664
9.6262
893
79 74 49
712 121 957
29.8831
9.6298
894
79 92 36
714 516 984
29.8998
9-6334
89s
80 10 25
716 917 375
29.9166
9.6370
896
80 28 16
719 323 136
29-9333
9.6406
897
80 46 09
721 734 273
29.9500
9.6442
898
80 64 04
724 150 792
29.9666
9.6477
899
80 82 01
726 572 699
29-9833
9-6513
, 900
81 00 00
729 000 000
30.0000
9.6549
Number
Square.
Cube.
Square Root.
Cube Root.
901
90Z
9°3
904
90s
81 18 01
81 36 04
81 S4 09
81 72 16
81 90 25
731 432 701
733 870 808
736 3'4 327
738 763 264
741 217 625
30.0167
30-0333
30.0500
30.0666
30.0832
9.6585
9.6620
9.6656
9.6692
9.6727
906
907
908
909
910
82 08 36
82 26 49
82 44 64
82 62 81
82 81 00
743 677 416
746 142 643
748 613 312
7SI 089 429
753 571 000
30.0998
3aii64
30.1330
30.1496
30.1662
9.6763
9-6799
9.6834
9.6870
9.6905
9"
912
9'3
914
91s
82 99 21
83 17 44
83 35 69
83 S3 96
83 72 25
756 058 031
758 55° 82s
761 048 497
763 551 944
766 060 87s
30.1828
30.1993
30.2159
30.2324
30.2490
9.6941
9.6976
9.7012
9.7047
9.7082
9.6
917
9.8
919
920
83 90 56
84 08 89
84 27 24
84 4S 61
84 64 00
768 S7S 296
771 09s 213
773 620 632
776 iS» 559
778 688 000
30.2655
30.2820
30.2985
30-3150
30-3315
9.7118
9-7153
9.7188
9.7224
9-7259
921
922
923
924
92s
84 82 41
85 00 84
85 "9 29
85 37 76
85 56 25
781 229 9C1
783 777 448
786 330 467
788 889 024
791 453 125
30.3480
30-3645
30.3809
30-3974
30.4138
9.7294
9-7329
9.7364
9.7400
9-7435
926
927
. 928
929
930
8s 74 76
55 93 29
86 II 84
86 30 41
86 49 00
794 022 776
796 597 983
799 178 752
801 765 089
804 357 000
30.4302
30-4467
30.4631
30.4795
30-4959
9.7470
9-7505
9.7540
9-7575
9.7610
93'
932
933
934
935
86 67 61
86 86 24
87 04 89
87 23 56
87 42 25
806 954 491
809 557 568
812 166 237
814 780 504
817 400 375
30.5123
30-5287
30.5450
30.5614
30.5778
9-7645
9.7680
9-7715
9.7750
9.7785
936
937
938
939
940
87 60 96
87 79 69
87 98 44
88 17 21
88 36 00
820 025 856
822 656 953
825 293 672
827 936 019
830 584 000
30.5941
30.6105
30.6268
30.6431
30.6594
9.7819
9.7889
9.7924
9-7959
941
942
943
944
945
88 S4 81
88 73 64
88 92 49
89 n 36
89 30 zs
833 237 621
835 896 888
838 561 807
841 232 384
843 908 625
30.6757
30.6920
30.7083
30.7246
30.7409
9-7993
9.8028
9.8063
9.8097
9-8132
946
947
948
949
950
89 49 16
89 68 09
89 87 04
90 06 01
90 25 00
846 59° 536
849 278 123
851 971 392
854 670 349
857 375 000
30.7571
30.7734
30.7896
30.8058
308221 1
9.8167
9.8201
9.8236
9.8270
9.8^05
'Number
Square.
Cube.
Square Root.
Cube Root. '
95'
952
i 953
9S4
9SS
90 44 01
90 63 04
. 90 82 09
91 01 16
91 20 25
860 085 351
862 801 408
8;6s 523 177
868 250 664
8*70 983 875
. 30.8383
30.8545
30.8707
30.8869
30.9031
9.8339
9-8374
9.8408
9.8443
9.8477
956
957
958
959
960
9t 39 36
91 58 49
9t 77 64
91 96 81
92 16 00
873 722 816
876 467 493
879 217 912
881 974 079
884 736 000
30.9192
30.9354
30.9516
30.9677
30.9839
9.8511
9.8546
9.8580
9.8614
9.8648
961
962
1 9G3
! 9.64
• 9^
92 3 21
92 54 44
92 73 69
92 92 96
' 93 '2 25
887 503 681
8^0 277 128
893 056 347
895 841 344
898 632 125
31.0000
31.0161
31.0322
31.0483
31.0644
9.8683
9.8717
9.8751
9.8785
9.8819
966
967
968
969
970
93 3« 56
93 5° 89
93 70 24
93 89 61
94 09 P°
901 428 696
9=4 231 063
967 039 232
909 853 209
912 673 000
31.0805
31.0966
31.I127
31.1288
31.1448
9.8854
9.8888
9.8922
9.§956
9.8990
971
972
973
' 974
97 S
94 28 41
94, 47 84
94 <'7 29
94 86 76
95 06 is
9 1 5 498 6 1 1
918 330 048
921 1G7 317
924 01,0 424
926 859 375
31.1609
31.1769
31.1929
31.2090
31.2250
9.9024
9.9058
9.9092
9.9126
9.9160
976
977
978
979
980
95 25 76
95 45 29
95 64 84
95 84 41
96 04 00
929 714 176
932 574 833
9 5 44J 352
9 8 313 739
941 l;)2 000
31.2410
31.2570
31.2730
31.2890
31.3050
9.9194
9.9227
9.9261
9.9295
9.9329
981
982
983
984
9.85
96 23 61
96 43 24
96 62 89
96 82 56
97 02 25
944 076 141
946 966 168
949 862 087
952 763 904
9-5 671 625
31.3209
31-3369
31.3528
31.3688
3<.3847
9.9363
9.9396
9-9430
9.9464
9.9497
986
987
988
989
99°
97 21 96
97 41 69
97 61 44
97 81 21
98 01 00
958 585 256
961 504 803
964 430 272
967 361 669
970 299 000
31.4006
31.4166
31.4325
31.4484
31.4643
9.9531
9-9565
9.9598
9.9666
991
992
993
994
995
98 20 81
98 40 64
98 60 49
98 80 36
99 00 25
973 242 271
976 191 488
979 '46 657
982 107 784
985 074 875
31.4802
31.4960
3i.5'i9
31.5278
31.5436
9.9699
9.9733
9.9766
9.9800
9.9833
996
997
998
999
1000
99 20 16
99 40 09
99 60 04
99 80 01
I 00 00 00
988 047 936
991 026 973
994 on 992
997 002 999
I oco 000 000
31.5595
31.5753
31.59"
31.6070
31.6228
9.9866
9.9900
9-9933
9.9967
10.0000