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SCIENTIA
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LIBRARY
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UNIVERSITY OF MICHIGAN
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Euclidis
Τ Η Ε
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T S
1
OF
E U C
C L I
C
D.
In which the Propoſitions are demonſtrated in a new and
forter'Manner than in former Tranſlations, and the
Arrangement of many of them altered,
To which are annexed
Plain and Spherical Trigonometry, Tables of Logaritlims from
I to 10000, and Tables of Sines, Tangents, and Secants, both
Natural and Artificial,
BY
GEORGE DOUGLAS,
Teaciier of Mathematics in the Academy at dyr.
L O N D ON:
Printed for 1. MURRAY, No. 32. Fleetſtreet; and C. ELLIOT,
Parlianient-ſquare, Edinburgh.
M.DCC.LXXVI,
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SION COLLEGE
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SOLD BY ORDER OF THL
PRBSIDENT AND GOVERNORS 1938.
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Τ Ο
MATTHEW STUART, D.D.
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Τ Η Ε
FOLLOWING EDITION
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Ο F Τ Η Ε
ELEMENTS OF
EU O L ID
A R E INSCR I B E D
BY
HIS MOST OBEDIENT
1
AND MOST HUMBLE SERVANT
GEORGE DOUGL A S.
!
lib.2071115
Hedger
2-5:39
37679
P R E F A C E.
Р E F
M
OST authors, from a natural anxiety to render their ſub-
jects as compleat as pollible, are in danger of being be-
trayed into prolixity: An attention to minute circumſtances may
be neceſſary in ſome kinds of compoſition, but prolixity is alto-
gether inexcuſable in a ſcientific writer. His object is to explain
the principles of ſcience in the moſt ſimple and perſpicuous
manner. To accompliſh this end, every fuperfiuity of language
and reaſoning ought to be ſtrictly guarded agairit. Whoever
has attended to books of ſcience will readily allow, that moit of
them are capable of abridgement; and that this abridgement,
inſtead of obſcuring, or rendering the ſubject more difficult,
will make it more clear and intelligible to the generality of itu-
dents.
Simplicity and conciſeneſs are peculiarly neceſſary in com-
municating the Elements of ſcience, which are always leſs in-
tereſting to the ſtudent than the practical parts. If the author
be tedious in this article, the mind, being entirely unacquainted
orn with the utility or application of elementary truths, is apt to re-
volt and abandon the ſtudy. But fimplicity and conciſeneſs are
more indiſpenſible in the elements of mathematics than any o-
ther ſcience. Unfortunately, however, too little attention has
hitherto been given to this circumſtance.
Euclid, an author long and juftly admired for the excellency
of his general method, has often gone ſo minutely to work in
his demonſtrations, as to render many plain propofitions not only
tedious, but difficult. His manner of demonſtrating is un-
queſtionably the beſt that has yet appeared, and therefore ought
to be followed : But it is by no means impoſlible to make his de
monſtrations as plain in much fewer words, and even to arrange
many of them in a different manner, without doing the leaſt
injury to his principles.
This taſk I have undertaken in the following ſheets. If
I have ſucceeded, one capital objection to the fudy of mathe-
matics is happily removed, as the Blements of Euclid may now
be learned in one half of the uſual time, and with greater eale
to the ſtudent.
That the reader may be the better prepared for the alterations
he may meet with, I have here mentioned a few, with the rea-
fons which induced me to make them.
Book
viii
PRE FACE.
Book I. ax. 10.“Two right lines do not bound a fgure;” in
ſtead of “ include a ſpace," the boundaries of ſpace, being diſ-
puted by metaphyſical writers, become unfit for a mathematical
axiom. Prop. 5. which is rather too tedious, I have proved from
prop. 4. in very few words, and have not uſed more freedom tham
is done in the demonſtration as it now ſtands. The ſecond part,
viz. the angles below the baſe, I have left out till the 13th is pro-
ved, from which it eaſily follows; and likewiſe in proving the
bafes equal in the 4th, I have changed the indirect proof, and
given a direct one, by which it is both ſhorter and eaſier com-
prehended. The manner in which I have enunced the 7th
prop. renders the ſecond part of the 5th unneceſſary; yet have
luppoſed no more given than what muſt be ſuppoſed before a
proof can be begun. But, thoſe who think it ought to be in
more general terms, I have indulged in the 21ſt, from which it
naturally follows. As ſome have thought axiom 12. not ſelf-
evident, and therefore ought not to be an axiom, I have added
a cor. to prop. 17. that convincingly proves it. The 35th and
37th are joined in one, as nothing can follow more naturally
than, if the wholes are equal, their halfs are likewiſe ſo. The
ſame may be ſaid of the 36th and 38th ; nor is it leſs natural to
prove it from half the parallelogram than to double the triangle,
and then take its half. I cannot agree with Mr Simpſon in
leaving out the corollaries from prop. 32. nor can I find any
reaſon for his fo doing.
Book II. I have varied the enunciation of ſeveral of the pro-
poſitions, and expreffed them in clearer terms. In the Sih pro-
poſition, the equality of the ſquares is proved in a ſhorter but
clearer manner than that preſently uſed. The 13th is retained
much in the ſame manner as in Commandine's Euclid; for,
though it be true of every ſide of a triangle ſubtending an acute
angle; yet, as the demonftration is general, and the perpendi-
cular falling within or without the triangle, makes no real alte-
ration, proving it in different figures becomes unneceſſary.
Book III. The first definition is challenged by Mr Simpſon,
which, he ſays, ought to be proved; for this I can fee no reaſon,
or any neceſſity of a proof, as the equality of coincident figures
is admitted, ax. 3. Book I. I have taken another demonſtra-
tion in place of that uſed in the ad propofition, which I thouglit
as mathematical as that uſed either by Commandine or Simpſon,
and much ſhorter. To the 8th prop. I have added, “ that only two
equal lines can fall either upon the convex or concave part of
" the circumference :" but the demonitration of the whole is
”
ſhorter than that preſently uſed. In the 16th,“ the angle of a
“ ſemicircle” is omitted, becauſe it follows more naturally as a
corollary. The 18th and 19th are joined in one, for the rea-
Lons already given. I have put a ſhort and natural demonſtra-
tion
;
1
;
2
6
P R E FACE.
ix
ܪ
tion in place of the 2d part of prop. 21. and changed the figure.
The 25th is ſhortened, and the 28th and 29th joined in one. In
the 31ſt,“ the angle of a ſegment is left out, but reſumed in
”
the cor. as it follows naturally from the propoſition. I have add-
ed a cor. to prop 37. which is found neceſſary in practice.
Book IV. is much ſhortened, the 12th, 13th, and 15th, are
demonſtrated in a different manner.
Book V. is ſhortened almoſt in every propoſition..
In Book VI. I have added a few words to the 5th def. which
renders it compleat ; the lemma added to prop. 22. is therefore
unneceſſary; as alſo def. A. inſerted aſter def. 11. book V. by
Mr Simpſon. The 5th and 6th propoſitions are joined in
one, as alſo the 14th and 15th; the demonſtrations are in ge-
neral ſhorter.
Book XI. Def. 10. is retained, as univerſally true, for the
reaſons given in the note at the end of the preface. Prop. 7. As
this propoſition has no dependence on any of the preceding
propofitions of this book, I have put it in place of the 6th, and
joined the 6th and 8th in one, by which the propoſition is made
both ſhorter and plainer than when ſeparate. The greateſt
part of the propoſitions of this book are conſiderably ſhortened.
Book XII. Prop. 5.and 6. are joined in one, and much ſhorten-
ed, and the demonſtrations in part new. The 8th and oth are
demonſtrated in a much ſhorter and more familiar manner;
the greateſt part of the roth and 11th being only a repetition of
the 2d, that Prop. is oniy referred to, as it is not neceſſary to de-
monſtrate a prop. twice over, nor has Euclid done ſo any where
at fo great length as in this book.
In PLAIN TRIGONOMETRY I have not inſerted any thiris
that depends for illuſtration on infinite ſeries, that being a ſub-
ject more proper for the higher parts of mathematics; but have
rendered the elements ſhort and comprehenſive, ſo as fully to
contain the principles of trigonometry, as well as to explain the
nature and uſe of the logarithmic canon.
In SPHERICAL TRIGONOMETRY, the propofitions are de.
monſtrated in a ſhort and eaſy method, from the principles of
plain trigonometry. The obſervations made on them by Mr
Cunn are left out, being wholly contained in the propoſitions,
and what he intends by them eaſily diſcovered in practice.
I have added a ſhort explanation, of the nature and uſe of
Sines, Tangents, Secants, and verſed Sines, both natural and
artificial ; and how to change Briggs's Logarithms to the Hyper-
bolic, and vice verſa, will examples of the above. To which
are annexed TABLES of the Logarithms of Numbers, of Sines,
Tangents, and Secants, both natural and artificial, which will
work to the fame exactneſs, of any extant, even to fecond and
third ininutes, or farther, if thought necedury
Upon
;
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M
Ꮲ A
P R E F A CE.
>
:
e
Upon the whole, although the above alterations are intended
to render the elements ealier and tooner acquired, yet are
not intended to indulge the indolence of either maſter or ftu.
dent. The Elements of Geometry being of ſuch extenſive ufe,
that a thorough knowledge of them is abſolutely neceflar
whether in the literary or mechanic profeſſion; the conciſeneſs of
;
the reaſoning, and conclufiveneſs of the argun:ents, render that
knowledge a neceſſary qualification for the pulpit or bar; and
in proſecuting the ſciences, this knowledge becomes abſolutely
neceffary : but the ſooner it can be acquired, a thorough know-
ledge of it may more eaſily be attained : and what is reſerved
of that time, which even an experienced Teacher would formerly
have taken up in barely demonſtrating the propoſitions, may be
employed in pointing out their particular beauties, the accura-
cy of the reaſoning, their uſe in the affairs of life, and their ap.
plication to the ſciences, which will be of great advantage to the
ſtudent, as he is hereby let into the beauties of the ſcience by
the time he formerly could have had but even a tolerable
knowledge of the method of demonſtration.
The author does not hereby mean to inſinuate, that this work
is without exception; that notwithſtanding the pains he has ta-
ken to render it as correct as poſſible, yet ſeveral inaccuracies,
both in the language and demonſtrations, may have eſcaped his
notice, which he hopes the learned will excufe, and lend their
aſſiſtance to render it more uſeful, if they fhall think it worthy
of another impreſſion,
That Mr Simpſon has fallen into a miſtake, in the demon-
ſtration he has given to prove the falfity of def. 10. Book XI.
will appear from the following obſervations :
He has proved that the triangles EAB, EBC, ECA, contain-
ing the one folid, are equal and fimilar to the three triangles
FAB, FBC, FCA, containing the other ſolid, and having the
common baſe ABC; he does not deny the equality of theſe fo-
Jids, but compares them with another ſolid contained by three
triangles GAB, GBC, GCA, and common baſe ABC, which
three triangles he neither proves equal nor ſimilar; but concludes,
that the folid contained by the three triangles GAB, GDC,
GCA, is not equal to the folid contained by the three triangles
EAB, EBC, ECA, and common baſe ABC, becauſe the one
contains the other. If he had proved, that the triangles GAB,
GBC, GCA, were equal and Gimilar to the other three triangles
EAB, EBC, ECA, and common baſe ABC, and then proved
the folids not equal, he would then have gained his point; but
as he has not even ſo much as attempted this, def. 10. muſt be
held as univerſally true ; at leaſt till ſome better argument is
produced againſt it.
But
P R E F A CE.
xi
But as he ſuppoſes it proved not univerſally true, he preſents
us with prop. A, B, C, after prop. 23. Book XI. to ſupply its
detect.
prop.
C. “ Solid figures contained by the ſame num-
“ ber of equal and ſimilar planes alike ſituated, and having none
“ of their folid angles contained by more than three plane angles,
are equal and ſimilar to one another.” But this prop. C. will e-
vidently appear inſufficient to ſupply this ſuppoſed defect, on ac-
count of the limited ſenfe in which it is taken; for, if ſolid fi-
gures, bounded by an equal number of equal and fimilar planes,
are not equal and ſimilar, but under this limitation, then prop.
15. Book V. muſt not be univerſally true, which I ſuppoſe will
not ealily be admitted ; and, if not admitted, then prop. C muſt
'
.
C
be a very infufficient foundation for proof of the following pro-
poſitions depending on it, viz. Prop. 25. 26. and 28. and con-
fequently eight others, viz. 27th, 31ſt, 32d, 330, 34th, 36th,
37th, and 4cth. Book Xl. all which are by this author toſſed off
their baſe, which is univerſally true, and placed upon this limit-
ed one.
;
Mr Simpſon farther objects, that though this definition be
true, yet ought not to be a definition, but a propoſition, and the
truth of it proved.
The ſame objection might be made with equal propriety to
ſeveral others; for example, why not prove the equality of theſe
angles which determine the equal inclination of planes, Def. 7.
Book XI. ar.d the equality of right lines equally diſtant from the
center, both which we may conclude to be Euclid's, as Mr
Simpſon does not object to them; for he would make us believe
none are Euclid's that he does not affirm to be fo, and that fre-
quently without any other reaſon given for it, but his own ipfe
,
dixit. If we conſider the nature of a definition, it is, if I mii.
take not, diftinguiſhing bodies from one another, by ſuch pro-
perties as cannot be applied to any other bodies, but thoſe it is
intended to diſtinguish. In which ſenſe, if the properties given
in this definition are ſuch as diſtinguiſh fimiiar and equal bo-
dies from others that are not ſo in every inſtance, then it is cer.
tainly a proper definition; but Euclid has ſometimes thought
proper to prove his definitions ; for example, def. 4. Book I'll.
which he has proved, prop. 14. of that book. This, it would ap-
pear, he has not thought neceſſary to prove, probably, if we may
be allowed to allign a reaſon in his name, that he has thought it
ſo ſelf-evident, that none would ever call the truth of it in que-
fion; but as the truth of it has been called in queſtion, the de-
finition may be proved in the following manner from Mr Simp-
fon's demonſtration to prove the contrary; for which obſerve his
own figure and demonſtration. He has proved the three triangles
EAB, EBC, ECA, containing the one folid, equal and ſimilar
to the three triangles FAB, FBC, FCA, containing the other fo-
lid
i
X18
PRE FACE.
G
E
lid, having the common baſe, ABC ; then, if the ſolid EABC is
not equal to the ſolid FABC, let it be equal to ſome ſolid as GA
BC, either greater or leſs than EABC, which cannot be; for
the one would contain the other; and if the ſolid angle is con-
tained by more than
three plane angles, e.
qual and ſimilar to one
another, then it can
E
be divided into angles
which are contained
by three equal and fi-
milar plane angles,
by Prop. 20. Book VI.
DI
and parts have the
ſame proportion as
B
C their like multiples,
by Prop. 15. Book V.
wherefore univerſally,
figures bounded by
F
an equal number of
equal and ſimilar planes are equal and fimilar.
N. B. In the references, when the propoſition referred to is
in the fame book with the propofition to be proved, the book is
not named, but only the number of the propoſition, but, if in
other book, both are named.
any
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Τ Η Ε
Ε L Ε Μ Ε Ν Τ S
M EN N T S
T
OF
E U C
C
L I D.
воок І.
Β
DEFINITION S.
A
Book I.
a
I.
Point is that which hath no parts or magnitude
II.
A line is length without breadth.
III.
The bounds of a line are points:
IV.
A right line is that which lieth evenly between its points.
V.
A ſuperficies is that which hath only length and breadth.
VI.
The bounds of a ſuperficies are lines.
VII.
A plain ſuperficies is that which lieth cvenly between its lines.
VIII.
A plain angle is the inclination of two lines to one another in
the ſame plain, which touch each other, but do not lie in the
ſame right line.
IX.
If the lines containing the angle be right ones, then the angle is
called a right-lined angle.
X.
When one right line ſtanding on another right line makes the
angles on each ſide thereof equal to one another, each of theſe
angles is a right one, and that line which ſtands upon the O-
ther is called a perpendicular to that whereon it ſtands.
A
XI. An
So wie gry on 2!:?
2
THE ELEMENTS
W
a
Book I.
XI.
An obtuſe angle is that which is greater than a right one.
XII.
An acute angle is that which is leſs than a right one.
XIII.
A term, or bound, is the extreme of any thing.
XIV.
A figure is that which is contained under one or more terms.
XV.
A circle is a plain figure hounded by one line, called the circum-
ference, to which all right lines drawn from a certain point
within the ſame are equal.
XVI.
That point is called the center of the circle.
XVII.
The diameter of a circle is a right line drawn through the cen-
ter, and terminated on both ends by the circumfererice, and
divides the circle into two equal parts.
XVIIT.
A ſemicircle is a figure contained under any diameter, and the
circumference cut off by that diameter.
XIX.
A ſegment of a circle is a figure contained under a right line,
and circumference cut off by that right line.
XX.
Right-lined figures are ſuch as are contained by right lines.
XXI.
Three ſided figures are ſuch as are contained by three lines.
XXII.
Four ſided figures are ſuch as are contained by four lines.
XXIII.
Many fided figures are ſuch as are contained by more than four
lines.
XXIV.
An equilateral triangle is that which hath three cqual ſides.
XXV.
An iſoſceles triangle, that which hath two ſides equal.
XXVI.
A ſcalene triangle, that which hath all the three fides unequal.
XXVII,
A right angled triangle is that which hath ene right angle in it.
XXVIII.
An obtufe angled one, that which hath one obtufe angle in it.
XXIX.
An acute angled triangle is that which hath all the angles leſs
than right ones.
XXX. A
OF EUCLI D.
3
اکره
XXX.
Book I.
A ſquare is that which hath four equal fides, and its angles all
right ones.
XXXI.
An oblong, or rectangle, is longer than broad, its oppoſite ſides
are equal, and its angles all right ones.
XXXII.
A rhombus, that which hath four equal ſides, but not right
angles.
XXXIII.
A rhomboides, whoſe oppoſite fides and angles are equal.
XXXIV.
All quadrilateral figures beſide theſe are called trapezia.
XXXV.
Parallel right lines are ſuch as, being produced both ways in
the ſame plain, never meet.
XXXVI.
A parallelogram is a figure whoſe oppoſite fides are parallel.
POS T U L A T E S.
G :
I.
TRANT that a right line may be drawn from any one
point to another :
II.
That a finite right line may be continued directly forwards: And,
III.
That a circle may be deſcribed about any center, with any
diſtance.
а
A X I 0 MS.
T
I.
HINGS equal to one and the ſame thing are equal to
one another.
II.
If equal things are added to equal things, the wholes will be ea
qual.
III.
If from equal things equal things be taken, the remainders will
be equal
.
If to unequal things equal things are added, the whole will be
unequal.
V. If
IV.
本
​E
THE ELEMENTS
1
Boox I
T
V.
I fron unequal things equal parts are taken, the remainders will
be une jual.
VI.
Things which are double one and the fame thing are equal be-
tween themſelves.
VII.
Things which are half one and the ſame thing are equal between
themſelves.
VIII.
Things which mutually agree together are equal to one another.
IX.
Any whole is greater than its part. .
X.
Two right lines do not bound a figure.
XI.
All right angles are equal to one another.
XII.
If a right line fall upon two right lines, making the inward
angles on the ſame fide leſs than two right angles, theſe right
lines continually produced will at laſt meet one another on
that ſide where the angles are leſs than right ones.
N. B. Any angle is expreſſed by three letters, of which
.
that at the vertex is named betwixt the other two.
PR 0.
I
OF EUCLID.
5
Book I.
PROPOSITION I. PROBLE M.
T
o deſcribe an equilateral triangle upon a given right
line.
a
a
3.
Let AB be the given right line, upon which it is required to
deſcribe an equilateral triangle.
About the center A, with the diſtance AB, deſcribe the circle
BCD *; and about the center B, with the diſtance BA, deſcribe a Poftulate
the circle ACE"; from the point C, where the two circles inter-
ſect each other, draw the lines CA, CB 6.
b Poft. I.
Then, becauſe A is the center of the circle DBC,AC is equal
to ABC, and becauſe B is the center of the circle ACE; BC c Definitie
c
is equal to BAC, but CA is proved equal to AB, and BC to AB; on 15.
therefore BC is equal to AC d: Thereforethe three fides AB,
BC, CA, are equal to one another: Therefore, upon the given
right line AB, there is deſcribed an equilateral triangle ABC:
Which was required.
dAxiom I.
PRO P. II. PROB.
A righe
T a given point to put a right line equal to a given
right line.
a
6
с
I.
Let A be the given point, and BC the given right line, it is
required to put a right line at the point A, equal to the given
right line BC.
With the center C and diſtance BC, deſcribe the circle BGH”;a Poſt. 3.
join AC", upon which deſcribe an equilateral triangle DAC, Poft. s.
produce DC that paſſes through the center to G, in the circum-
ference; and DA to any diſtance E d; with the center D, and d Poft. 2,
diſtance DG, deſcribe the circle KGL.
Then, becauſe C is the center of the circle BGH, BCis equal
to CG °; and becauſe D is the center of the circle KGL, DG e Def. 15.
is equal to DL º, but DC is equal to DAC: Therefore the re-
mainders AL, and CG, are equalf ; but BC, AL, are each f Ax. z.
proved equal to CG; and therefore equal to one anothers: 8 Ax. Ju
Therefore, from the point A, there is drawn the right line AL,
equal to the given right line BC: Which was required.
e
r
PROP.
6
TH E E L E M E N T S
Book I.
PRO P. III. PROB.
T"
WO unequal right lines being given, to cut off from the
greater a part equal to the lejler.
>
Required to cut off from the greater AB a part AE, equal to
the lefſer C.
From the point A draw a right line AD equal to Ca, about
Polt. 3. the center A, with the diſtance AD, deſcribe a circle DEFb;
.
then, becauſe A is the center of the circle DEF, AE is equal to
c Def. 15. ADC, but AD'is equal to C : Therefore AE is likewiſe equal
Ax. . to Cd: Which was required.
PRO P. IV. THE O R E M.
F there be two triangles having two ſides of the one cqual to
two ſides of the other, each to each ; and the angle contained
by the two ſides of the one, equal to the angle contained by the cor-
reſpondent ſides of the other; then the baſe of the one triangle will
be equal to the baſe of the other; the two triangles will be equal,
and the remaining angles of the one equal to the remaining angles
of the other, each to each, which the equal ſides fubtend.
Let the two triangles be ABC, DEF, having the two fides
AB, AC, equal to the two ſides DE, DF, each to each ; that
is, AB equal to DE, and AC equal to DF, and the angle BAC
equal to EDF; then the baſes BC_and EF will be equal, the
triangle ABC equal to the triangle DEF, the angle ABC equal
to the angle DEF; and ACB equal to DFE, each to each,
which the equal fides fubtend.
For, let the triangle ABC be applied to the triangle DEF, ſo
as the point A may coincide with the point D, the right line
AB with DE, then the point B will coincide with the point E;
for AB and DE are equal ; and, becauſe the angles BAC EDF
are equal, the right line AC will coincide with DF, and the
point C with F; for AC and DF are equal: Then, becauſe the
point B coincides with the point E, and C with F, the right
a Deffo lines BC and ET, will coincide“; and therefore equalb: There-
Ax, . fore the triangles ABC, DEF, are equal, the angle ABC e-
qual to the angle DET, and ACB to DFE, each to each, which
the equal fides ſubtend. Wherefore, &c.
1
PROP.
OF EUCLID.
了
​PRO P. V. THE O R.
Book I.
T.
H E angles above the baſe of every iſoſceles triangle are e-
qual to one another.
Let the triangle ABC be the iſoſceles triangle, having the ſide
AB equal to the ſide AC, then the angle ABC will be equal to
the angle ACB.
For, let a triangle DEF be likewiſe given, having the ſides
DE, DF, equal to the ſides AB, AC, each to each; and the
angle BAC equal to the angle EDF, then the baſes LF, BC,
are equal; and the angle ABC equal to the angle DEF,
and ACB to DFE, each to each ; which the equal lides ſub-
tenda ; but the fides AB, DF, are equal b; therefore the angles
,
ACB, DEF, are likewiſe equal; but the angles DEF, ABC,
are
equal b; therefore the angles ABC, ACB, are likewiſe equalb.
Wherefore, &c.
CORROLLARY. If any triangle is equilateral, it will likewiſe
be equiangular.
>
b
2 4.
L. AX, I,
b
PRO P. VI. THEO R.
I
F any triangle have two angles in it equal to one another, the
fides ſubtending theſe angles will likewiſe be equal.
و
A
Let the triangle ABC have the angles ABC, ACB, equal;
then the ſides AB, AC, will likewiſe be equal to one another.
For, if AB is not equal to AC, let one of them, as AB, be the
2
greater; from which cut off DB equal to AC“, and join DC;
then, becauſe DB is equal to AC, the two ſides DB, BC, are
equal to the two ſides AC, BC, and the angles ACB, DBC, e-
qual ; therefore the baſe AB is equal to the baſe DC, and the
triangles ABC, DBC, equal\; that is, a part equal to the whole; b fi
which is abfurd : Wherefore the fides AB, AC, are equal ; that
is, the triangle is iſoſceles. Wherefore, &c.
COR. Hence every equiangular triangle is alſo equilateral.
PRO P. VII. THEO R.
Τ
F from the extremity of any right line, io two different points
on the ſame ſide, there be dracun two righi lines equal to cue
another, the lines drau'n from the other extremity, io ile ſame
points, cannot be equal to one another.
I
If from the extremity A, of the right line AB, to the points
C,D, on the ſame fide, the two lines AD, AC, are drawn equal
to
3
Τ Η Ε Ε Ι Ε Μ Ε Ν Τ S
9
a
a S.
>
Book. I. to one another, from the extremity B, to the fame points C, D,
the lines BC, BD, are not equal to one another.
Join LC; for, becauſe AC, AD, are equal, the angles ADE,
ACD are equal a ; but the angle BDC is greater than ADCb, or
b Ax. 9. ACD, and much greater than BCD ; but, if BD is equal to
BC, the angles BCD, BDC, are equal, and likewiſe greater ;
which is impollible: Therefore, BD is not equal to BC. IfBD
is made equal to BC, it is proved in the fame manner that AC
is not equal to AD. Wherefore, &c.
. 9
1
TI
PRO P. VIII. T H E O R.
F two triangles have two ſides of the one, equal to two ſides of
the other, and the baſe of the one equal to the baſe of the other,
then the angles that theſe equal baſes ſubtend will be equal to one
anoi her.
Let the two triangles be ABC, DEF, having the two fides
AB, AC, equal to the two ſides DE, DF, each to each ; and the
bafes BC, EF, equal ; then the angles BAC, EDF, will be e-
qual.
For, let the triangle ABC be applied to the triangle DEF, ſo
that the right line BC may coincide with EF; then the point B
will coincide with the point E, and C with F; for the ſides BC,
EF, are equal : And the ſides BA, AC, will coincide with ED,
DF, and the point A with D. If not, let the point A fall in G;
then EG, ED, are equal ; for each are equal to BA “; and FG,
FD, will likewiſe be equal, which is impoſſible b. Wherefore
the point A cannot fall in G, and, for the ſame reaſon, in 110
point but D; therefore the angle BAC is equal to the angle
EDF. Wherefore, &c.
a.
a Ax. I.
b go
PRO P. IX. PROB.
TO
O cut a given right lined angle into two equal parts.
a
a 3
bi.
>
Let BAC be the given right lined angle required to be cut
into two equal parts.
Aſſume any point D, in the right line AB, and cut off AE e-
qual to AD *; join DE, upon which deſcribe the equilateral
triangle DEFb, and join AF; then the right line AF will
biſect the given angle BAC; for, becauſe DA is equal to EA,
and AF is common, and the baſe DF equal to EFb"; therefore
the angle DAF is equal to the angle EAFC: Therefore the
angle BAC is bifected by the right line AF. Which was re-
quired.
PROP
ce
OF EUCLI D.
$
9
BOOK I.
PRO P. X, PRO B.
TO
O cut a given finite right line into two equal parts.
a
Let AB be a given right line, required to be cut into two e-
qual parts ; upon it deſcribe an equilateral triangle ABC; bi-
ſect the angle ACB by the right line CD a; then is the right 9.
line AB biſected in D.
For, becauſe AC is equal to CB, and CD common, and the
angles ACD, BCD, equal", the bafes AD, BD, are equal b: b 4.
Therefore the right line AB is bifected in D. Which was reo
quired.
1
PRO P. XI. PROB.
Toin given in the same
O draw a line at right angles to a given right line from a
point given in the ſame.
a 3•
Let AB be the given right line, and C the given point in it,
from which it is required to draw a right line, at right angles to
the given right line AB.
Afſume any point D in AC, and make CE equal to CD ;
upon DE deſcribe an equilateral triangle DEFD, and join FC, b 1.
which will be at right angles to AB. For, becauſe DC, CE,
are equal", and CF common, the two ſides DC, CF, are equal c. conſt, .
to the two ſides EC, CF, and the baſes FD, FE, equal b, the
angles FCD, FCE, are likewiſe equald: Therefore each of them d 8.
is a right angle, and FC perpendicular to AB Which was e Def. 10
required.
1
€
PRO P. XII. PRO B.
T
10 draw a right line perpendicular to a given indefinite right
line from a given point out of it.
Let AB be the given right line, and C the point given out of
it, from which its required to let fall a perpendicular upon the
indefinite given right line AB.
Afſume any point D, on the oppoſite ſide of the right line
AB; about the center C, with the diſtance CD, deſcribe a circle
EDG ; bifect EG in H, join CG, CH, CE; then CH is the
perpendicular required.
B
For
IO
THE ELEMENTS
!
>
Book I. For GH, HC, are equal to EH, HC; and the baſes GC,
WEC , equal: Therefore the angles GHC, EHC, are equal 6.
a def. 15. Each of them are right angles, and HC perpendicular to AB.
Which was required.
9
.
b 8.
c def. 1o.
PRO P. XIII. THEO R.
HEN a right line ſtands upon a right line, making angles
with it, theſe angles are either two right angles, or, to-
gether, equal to two right angles.
WH,
>
a
a def 10.
3
b II.
Ca&. I.
For, let a right line, AB, ſtand upon the right line CD, ma-
king anglis CBA, ABD; theſe angles ſhall either be two right
angles, or, together, equal to two right angles.
For, if CBA, ABD, be equal, they are right angles a ; if not,
from the point B draw BE, at right angles, to DCV: Therefore
the two right angles CBE, EBD, are equal to the three angles
ABC, ABE, EBD; but the two angles ABC, ABD, are equal
to the fame three angles: Therefore the two angies ABC, ABD,
are equal to the two angles CBE, EBD; that is, equal to two
right anglesº. Wherefore, &c.
Cor. If the two ſides of an iſoſceles triangle ADE, be pro-
duced to B, C, the angles below the baſe will be equal to one a-
nother ; for the angles ADE, EDB*, are equal to two right
anglesd, and AED, DEC, are equal to two right anglesd, but
the angles ADE, ALT), above the baſe, are proved equal e :
Therefore the remaining angles LDL, DEC, are equal f: There-
fore, in every iſoſceles triangie, the angles above the baſe are e-
qual to one another; and, if the ſides be produced, the angles
below the baſe are likewiſe equal to one another.
*
* Fig, to
prop. 9.
;
d 13
3
е
e 5.
fax. 3.
:
PRO P. XIV. THE O R.
>
F to any right line, and point therein, two right lines be drawn
from oppoſite points, making the alljacent angles together e-
qual to two right angles, theſe two right lines will make oile conti-
nued right line.
>
For, if to any right line AB, and point B therein, be drawn
B
two right lines, CB, DB, from the oppoſite points, C, D, making
the angles ABC, ABD, equal to two right angles; then DBC
will be one right line. If not, let CBE be one right line; then
the angles ABC, AßE, will be equal to two right angles a; but
ABC, ABD, are equal to two right angles b; Therefore the
a
L. 13.
k. hyp,
two
OF EUCLI D.
/
II
2
two angles ABC, ABD, are equal to the two angles ABC, Book I.
ABE. Take ABC from both ; then the angle ABE will be
qual to the angle ABD, a part to the whole; which cannot be.
Wherefore, &c.
COR. Hence two right lines CBD, CBE, cannot have a com-
mon ſegment as CB; or BD, BE, cannot both be in a right
line with CB.
PRO P. XV. T H E O P.
IF
F two right lines mutually cut each other, the oppoſite angles
are equal.
a
>
Let the right lines AB, CD, mutually cut each other in the
point E, the angles AEC, DEB, will be equal; and likewiſe
the angles CEB, AED, equal to one another. For, becauſe the
right line CE falls upon the right line AB, the angles ALC,
CEB, are equal to two right anglesa. For the ſame reaſon the a 13.
angles AED, AEC, are equal to two right anglesa : Therefore
the two angles AEC, CEB, are equal to the two angles AEC,
AEDO. Take the common angle AEC from both, the remain- Ax. fo.
ing angles CEB, AED, are equal. Again, becauſe AEC,
"
AED, are equal to two right anglesa, and AED, DEB, equal to
,
two right angles, take the common angle AED from both, the
remaining angles AEC, DEB, will be equal. Wherefore,
&c.
Cor. 1. Hence, two right lines cutting each other, the angles
,
at the ſection are equal to four right angles.
2. All the angles conſtitute about any point are equal to four
right angles.
Ć AX, 3-
2
PRO P. XVI. THEO R.
Fone ſide of a triangle be produced, the outward angle will be
greater than either of the inward oppoſite angles.
a
Let ABC be a triangle, and one of its fides BC be produced
to D, the outward angle ACD will be greater than the angle
CBA, or BAC.
For, biſect AC in E a; join BE, which produce to F; make a 10,
EF equal to EB, and join FC ; then the two fides AE, EB, are
equal to the two fides FE,EC, and the angles AEB, FEC, e-
qual b; Therefore the baſes FC, AB, are equal; and the angles base
ECF,
b
12
Τ Η Ε Ε Ε Ε Μ Ε Ν Τ S
€ 4.
Book. I. ECF, EAB, likewiſe equal : Therefore the angle ACD is
greater than the angle BAC. In like manner, if the ſide
BC is biſected in E, EF made equal to AE, and FC join-
d 15. ed, the angle BCG, or ACD4, is greater than ABC; but
ACD is likewiſe proved greater than BAC. Wherefore,
&c.
P R 0 P. XVII, T H E 0 R.
T"
WO angles of any triangle, however taken, are, together,
leſs than two right angles.
á 16.
b 13
Let ABC be the triangle, any two angles in it are leſs than
two right angles.
For, produce BC both ways to D, E; then, becaufe the out-
'ward angle ACD is greater than ABC“, add ACB to both;
then the angles ACD, ACB, are greater than ABC, ACB, or,
BAC, ACB ; but ACD, ACB, are equal to two right angles b:
Therefore ABC, ACB, or ACB, BAC, are leſs than two
right angles. For the ſame reaſon, ABE, ABC, are greater
than BAC, ABC. Whercfore, &c.
Cor. Hence, if a right line fall- upon two right lines, making
the inward angles on the ſame fidë leſs than two right angles,
theſe lines will meet one another on that ſide where the angles
are leſs than right ones.
14
PRO P. XVIII. THEO R.
T
HE greater ſide of every triangle ſubtends the greater
angle.
а 3.
b 16.
Let ABC be a triangle, and the ſide AC greater than AB ;
then the angle ABC will he greater than the angle ACB.
For, from the greater AC cut off. AD, equal to AB, join
DB ; then, becauſe ADB is greater than ACB b, ABD is like-
wiſe greater than ACB, and 4BC much greater. Wherefore,
&c.
و
PRO P. XIX. THEO R.
T greater fide.
HE greater angle of every triangle is fubtended by the
IN
1
OF EUCLID.
13
In the triangle ABC let the angle ABC be greater than the Book I.
angle BCA ; the fide AC will be greater than AB. If not, let
AC be either equal or leſs than AB. If equal, then the angle
ABC is equal to ACB * ; but it is not b: Therefore AC is not a 5.
.;
s
equal to AB. If AC is leſs than AB, the angle ABC is leſs
than ACBC; but it is not : Therefore AC is not leſs than AB. C 18.
It is therefore greater, ſince it has been proved neither equal
nor leſs. Wherefore, &c.
b Hyp.
с
PRO P. XX. T H E O R.
T"
W O fides of any triangle, however taken, are greater
than the third.
а
a .
In any triangle, ABC, two ſides of it, however taken, are
greater than the third, viz. AB, AC, greater than BC; AC,
BC, greater than AB; or BC, AB, greater than AC. For,
produce any fide, as BA, to D; make AD equal to AC a; and a 3.
join DC: Then, becauſe AD is equal to AC, the angles ADC,
ACD, are equalb; but the angle BCD is greater than ACD,
bs.
that is, than ADC: Therefore the ſide BD is greater than BCC;C 19.
but BD is equal to BA, AC: Therefore the ſides BA, AC, are
greater than BC. Wherefore, &c.
>
с .
PRO P. XXI. THEO R.
9
IR
F two right lines be drawn from the extreme points of one ſide
of a triangle, ito a point within the ſame, theſe two right lines
will be leſs than the ſides of the triangle, but contain a greater
angle.
3
From the extreme points of the right line BC, let the two
right lines BD, CD, be drawn to the point D, within the ſame;
theſe lines (hall be leſs than the ſides BA, AC; butthe angle BDC
will be greater than'BAC.
For, produce BD to E; then the two ſides BA, AE, are great-
er than the third fide BE a; add EC to both ; then BA, AČ, are a 20.
,
greater than BE, EC6. For the ſame reaſon, BE, EC, are b. Ax. 4:
greater than BD, DC; but BA, AC, are greater than BE,
ĒC; therefore much greater than BD, DC. But the angle
BDC is greater than BAC; for the angle BDC is greater than
BEC“; and BEC is greater than BAČ C: Therefore BDC is cles
much greater than BAC. Wherefore, &c.
COR.
1
14
THE E L E M E N TS
Book I.
COR. Hence BD, DC, are not equal to BA, AC, each to
each. Wherefore, if in any caſe it is thought neceſſary to prove
that part of Prop. VII. when the one point falls within the
triangle, it is evident from this.
PRO P. XXII. PROB.
T:
O make a triangle, whoſe ſides are equal to three given right
lines, if any two of theni, however taken, are greater than
the third.
Let A, B, C, be the three given right lines, any two of
which are greater than the third. Take any right line bounded
at D, but not bounded at E, from which cut off DF equal to A,
FG equal to B, and make GH equal to C; then, with the
center F, and diſtance DF, deſcribe the circle DKL; with the
center G, and diſtance GH, deſcribe the circle KLH ; from the
point K, where the circles cut each other, draw the right lines
Def. 15. FK, KG; then FD is equal to FK -; but FD is equal to A;
therefore FK is equal to A. For the ſame reaſon GK is equal
to C, and FG is equal to B: Therefore the three ſides FK, FG,
GK, of the triangle FKG, are equal to the three given right
lines, A, B, C. Wherefore there is conſtitute, &c.
3
PRO P. XXIII. PRO B.
r
A
Ta giton point, in any right line, to make an angle en
qualto a given right lined angle.
!
а
Let A be the given point in the right line AB; it is required
to make an angle equal to the right lined angle DCE.
Afſume any points D, E, in the right lines CD, CE, and join
DE. At the point A, in the line AB, make a triangle AFG, whoſe
fides are equal to the three right lines CD, CE, DE; then,
becauſe the two fides GA, AF, are equal to the two ſides CE,
CD, each to each, and the baſes GF, ED, equal, the angles
GAF, ECD, are equal b. Wherefore there is conftitute, &c.
ز
ho
PRO P. XXIV. THE O R.
I
F two triangles have two ſides of the one cqual to two ſides of
the cther, each to cach, and the angle contained by the two
fides of the one greater than the angle contained by the correſpond-
ent
&
OF EUCL Í D.
>
15
ent ſides of the other ; then the baſe that fubtends the greater Book I.
angle of the one triangle ſhall be greater than the baſe of the om n
ther.
2
I
b Axis
Let ABC, DEF, be the two triangles, having the two ſides
BA, AC, equal to the two fiies ED, DF, each to each, but
the angle BAC greater than EDT; then the baſe BC will be
greater than EF.
For, make the angle EDG equal to BAC, and DG to AC;
join EG ; then the baſes BC, EG, will be equal ?. Now, iſt, if a 4
the right line EF fall upon EG, then EG will be greater than
EF b; and therefore, BC greater than EF.
.2. If EF fall above EG, then F is a point within the triangle;
therefore the ſides DF, FE, are leſs than DG, GE®; but DG, 21.
DF, are equal d; therefore EG, or BC, is greater than EF,
3. If EF fall below EG, join FG; then DF, DG, are equald:
Therefore the angles DGF, DFG, are equal; and the whole f 5-
angle EFG greater than DFG, or GF, and much greater than
EGF6; but the greater angle is fubtended by the greater fides : & 33
Therefore EG or BC is greater than EF. Wherefore, Sc.
.
d AXI.
e AX. S.
>
f
.
P R 0 P. XXV. T H E 0 B.
IF
F two triangles have two ſides of the one equal to two ſides of
the other, each to each, and the baſe of the one greater than the
baſe of the other, the angle that the greater baſe ſubtends ſhall be
greater than the other.
Let the two triangles be ABC, DEF, having the fides AB,
AC, equal to the two ſides DE, DF, each to each, and the
baſe BC greater than the baſe EF; then the angle BAC will be
greater than the angle EDF. If not, it will be equal or leſs.
If equal, the baſes PC, EF, will be equala ; but they are not 0.74
If leſs, the bale BC will be leſs than EFC; but it is not
Therefore, ſince the angle BAC is neither equal nor leſs than
EDF, it muſt be greater. Wherefore, &c.
b
6 Hype
Element
C
PRO P. XXVI. THEOR
IF
F two triangles have two angles of the one equal to iwo angles
of the other, each to each, and a ſide of the one equal to a ſide of
the other, either the fide lying between the equal angles, or ſubtend-
ing
1
16
T H E E L E M E N T S
Book I. ing one of them, the remaining ſides of the one triangle will be equal
to the remaining ſides of the other, each to each, and the remain-
ing angle of the one equal to the remaining angle of the other.
છે
a 3.
>
b 4.
b
с Нур.
с
Let the two triangles be ABC, DEF, having the two angles
ABC, ACB, of the one, equal to DEF, DFE, of the other,
each to each,
1. Let the ſide BC be equal to EF, viz. the fides lying be-
tween the equal angles; then the fides BA, AC, will be equal
to the ſides ÉD, DF, each to each; and the angles BAC, EDF,
equal. For, if the fide AB be not equal to DE, let one of them,
as AB, be the greater; from which cut off GB equal to DE*,
join GC; then, ſince GB, BC, are equal to DE, EF, and the
angles GBC, DEF, equal, the baſes GC, DF, are equal; and
the angles GCB, DFE, equal b; but the angle DFE is equal
to ACổ C: Therefore GCB is equal to ACB, a part to the
whole; which is impoſſible: Therefore GB is not equal to DE,
;
nor is any fide but AB equal to DE: Therefore AB, BC, are
equal to DE, EF; the angle ABC, to DEF; and the baſe AC
to DF
2. Let the fides AB, DE, which ſubtend the equal angles, be
equal; if any of the ſides, as BC, be not equal to EF, let BC
be the greater ; cut off BH equal to EF a , join AH; then, be-
cauſe AB, BH, are equal to DE, EF, and the angle ABH to
DEF, the baſe AH equal to DF, and the angle AHB to
DFE); but the angle ACB is equal to DFEC: Therefore the
angle AHB is equal to ACB, and likewiſe greaterf; which is
impoſſible. Wherefore, &c.
t
d4
>
a
b
e Ax, 1.
fio.
1
PRO P. XXVII. T H E O R.
IF a right line fall upon two right lines, making the alternate
angles equal, theſe right lines will be parallel.
Let the right line EF fall upon the two right lines AB, CD,
making the angles AEF, EFD, equal, the right lines AB, CD,
will not meet one another, whether produced towards B, D, or
A, C. Let them be produced; and, if poſſible, meet in the
point G; then EGF is a triangle; the outward angle AEF is
greater than EGF, or EFG*: but AEF, EFG, are equalb,
and likewiſe greater ; which is impoſſible : Therefore AB, CD,
will not meet, if produced toward B, D. For the ſame reaſon
they will not meet, if produced toward A, C: Wherefore AB,
CD, are parallel
PROP
2
a 16.
b Hyp.
OF EUCLID17
D.
..
Book I.
PRO P. XXVIII. THE OR.
F a right line fall upon two right lines, making the outward
angle equal to the inwari and oppoſite, on the ſame ſide, or the
inward angles on the ſame fide equal to two right angles; theſe
two right lines Jhall be parallel.
a
Let the right line EF fall upon the two right lines AB, CD,
making the outward angle EGB equal to the inward and oppo-
fite GHD; or the inward angles BGH, GHD, together, equal
to two right angles; then AB, CD, will be parallel.
For, becauſe the angles EGB, GHD, are equal", AGH is a hyp.
equal to EGBb; and therefore equal to GHDC: therefore A.B 15.
is parallel to CD 4. Again, becauſe the angles BGH, GHD, are c Ax. I.
equal to two right onese; but AGH,BGH, are equal to two right
angles €; therefore AGH, BGH, are equal to BGH, GHD .e 13,
Take the common angle BGH from both, the remainders AGH,
GHD, are equalf; but theſe are alternate angles: Therefore { Ax. 3.
f .
AB is parallel to CD d. Wherefore, &c.
d 27.
e
с
PRO P. XXIX. THE OR.
T .
I F a right line fall upon two parallel lines, the alternate angles
will be equal; the outward angle equal to the inward and op-
poſite, on the ſame ſide ; and the two inward angles on the ſame
fide equal to two right angles.
2
For, let EF fall upon the two parallel lines AB, CD, the al-
ternate angles AGH, GHD, will be equal; the outward angle
EGB equal to the inward GHD; and the two inward angles
BGH, GHD, equal to two right angles.
For, if the angle AGH is not equal to GHD, let one of
them be greater, as AGH; then the right lines AB, CD, pro-
duced toward B, D, will meet one another in ſome point; but
they are parallel; therefore cannot meet 6 : Therefore AGH is
b
not greater than GHD. For the ſame reaſon it is not leſs;
therefore it is equal. But EGB is equal to AGH ; therefore c 15.
IGB is equal to GHD d. Add BGH to both; then EGB, d Ax. s.
BGH, are equal to BGH, GHD ; but EGB, BGH, are equal e Ax, 2.
to two right anglesf: Therefore BGH, GHD, are equal to two £ 13.
right angles. Wherefore, &c.
a Cor. 1g.
b Def. 35.
€
e
PROP.
18
Τ Η Ε Ε L Ε Μ Ε Ν Τ S
Book I.
PRO P. XXX. THE O R.
R
IGHT lines parallel to one and the ſame right line, are pa
rallel to one another.
>
2 29°
9
Let AB, CD, be two right lines, each parallel to EF; AB will
be para jel to CD.
Let GK fall upon them ; then, becauſe GK falls upon the
parallels AB, EF, the angles AGH, GHF, are equala. A-
gain, becauſe GK falls upon the parallels EF, CD, the outward
angle GHF is equal to the inward and oppoſite GKD; there-
fore the angles AGK, GKD, are each equal to GHF; therefore
equal to one another b: Therefore AB is parallel to CD
Wherefore, &c.
b AX, I.
C 27.
PRO P. XXXI. PRO B.
.
O araw a right line through a given point parallel to a given
T
right line.
-
r
It is required, through the point A, to draw a right line pa.
rallel to the right line BC.
Aſſume any point D, in BC; join AD; and make the angle
DAE equal to the angle ADCa; join EA, and produce it to
F; then the alternate angles EAD, ADC, are equal to one a-
nother 6: Therefore EF, BC, are parallel . Wherefore, &c.
a 23
a
b Conſt.
C 27.
PRO
P R O P. XXXII. THE O R.
1
IF
F one ſide of a triangle be produced, the outward angle is equal
to both the inward oppoſite angles ; and the three inward
angles are equal to two right angles.
Let ABC be a triangle, CD a fide produced; the outward
angle ACD is equal to the inward and oppoſite angles ABC,
BAC; and the three angles ACB, ABC, and BAC, are toge-
ther equal to two right angles. Through C draw CE parallel to
AB; then the angle BAC is equal to ACE* ; and the angle
ECD, to ABC^; therefore the whole angle ACD is equal to
the two angles ABC, BAC. Add the angle ACB to both; then
thetwo angles ACD, ACB, are equal to the thrce angles ABC,
ACB,
a
a 29
ز
a
>
>
OF EUCLI 19
CLIDD.
.
.
ACB, BACb; that is, equal to two right angles. Where- Book I.
fore, &c.
Cor. 1. Hence all the three angles of any one triangle are 5 Ax, I.
equal to all the three angles of any other triangle, either ſepa. C 13.
rately or taken together.
2. If two angles of one triangle be equal to two angles of a-
nother triangle, either ſeparately or together, the remaining
angle of the one is equal to the remaining angle of the o-
ther.
3. If one angle of a triangle be a right one, the other two
angles are together equal to a right angle.
4. If the angle included by the equal ſides of an iſoſceles
triangle be a right one, each of the other angles will be half a
right one.
5. Any angle in an equilateral triangle is one third of two
right angles, or two thirds of one right angle.
6. If one angle of a triangle be equal to the other two, that
angle is a right one ; for, if the ſide is produced, the adjacent
angle is equal to the other two ; therefore each of them are
right angles.
7. All the inward angles of any right lined figure make twice
as many right angles, abating four, as the figure has fides.
For any right lined figure can be divided into triangles, the in-
ward angles of each equal to two right angles, and all the
triangles together equal to the number of fides of the figure, a-
bating two : Therefore all the inward angles will be equal to
twice the number of fides, abating four.
8. All the outward angles of any right lined figure are equal
to four right angles. For all the outward and inward angles
together are equal to double the number of fides ; but the in-
ward angles are equal to double the number of fides, abating
four: Therefore the outward are equal to four right angles.
ܪ
PRO P. XXXIII. T H E O R.
I
Ftwo right lines join two equal and parallel right lines toward
the ſame part, theſe lines will be equal and parallel.
ܪ
Let AB, CD, be two equal and parallel right lines ; join
AC, BD; then will the right lines AC, BD, be equal and
parallel.
For, becauſe AB, CD, are parallel, and BC falls upon them,
the angle ABC is equal to BCD ; but, becauſe AB is equal to a 29.
CD, and BC common; and the angle ABC equal to BÇÕ, the
baſe AC is equal to BD, and the angle ACB equal to CBD b; b 4.
but
2
20
THE ELEMENTS
Book I. but theſe are alternate angles : Therefore AC is parallel to
BD', and likewife equal. Wherefore, &c.
a
, 29
PRO P. XXXIV. T H E O R.
HE oppoſite ſides and oppoſite angles of every parallelo-
gran are equal; and the diameter divides it into two
equal parts.
T
TA
1
29
Let ABCD be a parallelogram, the oppoſite fides AB,
,
CD; AC, 3D, are equal ; the angle CAB equal to BDC, and
ACD to ABD; and the diameter BC biſects it.
For, becauſe AB is parallel to CD, and BC falls upon them,
the angle ABC is equal to BCDa. For the ſame reaſon ACB
is equal to CBD; therefore the two angles ABC, ACB, in the
triangle ABC, are equal to the two angles CBD, BCD, in the
triangle BCD; and the ſide BC common to both: Therefore
the two ſides AC, AB, of the one triangle, are equal to the
two fides BD, DC, of the other, each to each; and the angle
BAC equal to BDC, and ACD to ABD b. Again, becauſe
the two ſides AC, AB, are equal to the two fides BD, DC,
each to each, and the angle CAB equal to BDC, the baſe BC
common: Therefore the triangles are equal"; and BC biſects
the parallelogram. Wherefore, &c.
b 26.
C4.
C 4.
C
PRO P. XXXV. and XXXVII. THEO R.
a 34,
Arallelograms and triangles, conſtitute upon the ſame baſe,
and between the ſame parallels, are equal between theni-
felves, viz. parallelogram to parallelogram, and triangle to
triangle.
Let ADCD, EBCF, be two parallelograms (Fig. 2.) conſtitute
,
upon the ſame baſe BC, and between the ſame parallels BC,
AF; the parallelograms ABCD, EBCF, are equal.
For, becauſe AD), ET, are each equal to BC”, they are equal
to one another 6. If the point E coincide with D. (Fig. 1.) each
of the parallelograms are double the triangle DBC; therefore e-
qual to one another. If AD is leſs than AE, add DE to
c
both ; then the whole AE is equal to DF, DC to AB, and the
angle FDC to LABf: Therefore the triangles FDC, EAB,
are equal 8. Take DGE from both; the trapeziums, ADGB,
FEGC, are equal h. Add the triangle GBC to both; then the
whole parallelogram ABCD is equal to the parallelogram
>
b Ax. I.
CAX. 6.
d
d AX, 2.
e 34.
f 29.
§ 4.
h
EBCF.
11 Ax. 3.
OF EUCLID.
21
EBCF d. If AD is greater than AE, take DE from both; then Book I.
the remainder AE will be equal to DF5, the triangle AEB to
DFC. Add EBCD to both, then the parallelograms ABCD, Ax. 2.
EBCF, are equald: So, likewife, if the diameters AC, BF, bé h Ax. 3.
drawn, then the triangle ABC will be equal to FBCi. Where- i 34. and
fore, &c.
Cor. Hence every parallelogram is equal to a right angled
parallelogram, conſtitute upon the ſame baſe, and between the
fame parallels; and every triangle conſtitute upon the ſame
baſe, and betwixt the ſame parallels, is half the rectangle.
AX. 7.
PRO P. XXXVI. and XXXVIII. THE OR.
P
Arallelograms and triangles, conftitute upon equal baſes, and
between the ſame parallels, are equal to one another, viz.
parallelogram to parallelogram, and triangle to triangle.
Let the parallelograms ABCD, EFGH, be conſtitute upon
the equal baſes BC, FG, and between the fame parallels AH,
BG ; the parallelogram ABCD will be equal to EFGH.
For, join EB, CH, the parallelograms AC, LG *, are each
equal to the parallelogram EC 4; therefore equal to one ano-a 35.
therb. Join AC, FH; then the triangles ABC, HGF, are e- ) AX. 1.
qual. Wherefore, &c.
C 34. and
AX. 7.
PRO P. XXXIX. THE O R.
QUAL triangles, conſtitute upon the ſame baſe, on the fanie
fide, are between the fame parallels.
>
,
Let the equal triangles ABC, DBC, be conſtitute upon the
fame baſe, BC, on the ſame ſide ; the right line AD, that
joins their vertex, will be parallel to BC.
If not, draw AE, parallel to BC; join EC; then the
triangles ABC, EBC, are equal a ; but DBC is equal to ABC"; a 35.
therefore DBC, EBC, are equal, a part to the whole; which b Hype
is impoflible. Therefore no line but AD is parallel to BC.
Wherefore, &c.
ز
PROP
* Parallelograms are expreſſed by the letters at the oppoſite angles,
THE ELEMENTS
Book I.
PRO P. XL. THEO R.
E
QUAL triangles, conſtitute upon equal baſes, on the ſame
fide, are between the ſame parallels.
;
Let ABC, DGE, be equal triangles, conſtitute upon the e-
qual baſes BC, GE, on the ſame lide; then AD is parallel to
BE. If not, draw AF parallel to BE; join FE ; then the
triangle ABC is equal to FGE; but DGE is equal to ABC b;
a
therefore DGE is equal to FGE, a part to the whole; which
is impollible : Therefore AD is parallel to BE. Wherefore,
&c.
а 3б,
b hyp.
b
و
PRO P. XLI. THEO R.
T
Fa parallelogram and triangle be conſtitute upon the ſame baſe,
and between the ſame parallels, the parallelogram will be
double the triangle.
Let the parallelogram be ABCD, and triangle EBC, having
the ſame baſe BC, and be between the fame parallels AE, BC,
the parallelogram ABCD is double the triangle EBC; join
AC.
Then the triangle ABC is equal to EBC?; but the paralle-
logram ABCD is double the triangle ABC b; and therefore
double EBC. Wherefore, &c.
a 35.
b 34.
PRO P. XLII. PROB.
T
conſtitute a parallelogran equal to a given triangle, ha-
ving an angle in it equal to a given right lined angle.
a
a 10,
و
by 23
C 31.
d 36.
Let the given triangle be ABC, and right lined angle D, it
is required to conſtitute a parallelogram equal to the given
triangle ABC, having an angle in it equal to D.
Bilect BC in E, make an angle CEF equal to Db; through
A draw AG parallel to CE; and through C draw CG parallel
to EF'; join AE; then the triangles ABE, AEC, are equal d;
and ABC is double AEC ; but the parallelogram EG is double
the triangle EACe: Therefore the parallelogram EG is e-
qual to the triangle ABCf, and the angle FEC equal to D.
Wherefore, &c.
C.
2
:
€ 41.
AX6.
PROP.
OF EU CL I V.
23
Book I.
P R O P. XLIII. THE O R.
I.
N every parallelogram, the complements that ſtand about the
diameter are equal to one another.
Let ABCD be a parallelogram ; BD its diameter; the parts
of which BK, KD, the diameters of the parallelograms HKFD,
EBGK; the remaining parallelograms AEKH, KGCF, its
complements, are equal to one another.
For, becauſe DB is the diameter of the parallelogram ABCD,
the triangles ADB, DBC, are equal a. For the ſame reaſon, a 34.
the triangle HKD is equal to DFK, and EBK to BKG; where-
fore the triangles HKD and EKB are equal to DFK, BKG, 6.b.Ax. z.
Take HKD, EKB, from ADB, and DFK, BKG, from DBC, Ax 3.
there remains AEKH equal to KGCF. Wherefore, &c.
>
PRO P. XLIV. PRO B.
0 apply a parallelogram to a given right line equal to a given
triangle, having an angle in it equal to a given right lined
angle.
1
r
a
ܪ
3
It is required, upon the given right line AB, to make a paral.
lelogram equal to a given triangle C, having an angle in it e-
qual to a given angle D.
Make the parallelogram FGBE equal to the triangle C, ha-
ving the angle EBG equal to D ; put BE in a right line with a 42,
AB; and produce FG to H ; through A draw AH parallel to
GB, or FE ; join HB. Now, becauſe the angles EFH, FHA,
are equal to two right angles, the angles EFH, FHB, are leſs
than two right angles; then FE, HB, being produced, will meet
in ſome point; which let be K; through which draw KL
parallel to FH; and produce GB, HA, to M, L; wherefore
FHLK is a parallelogram, whoſe diameter is HK ; and whoſe
complements FGBE, BALM, are equal d; but FGBE, was d 43.
made equal to C; and the angle EBG equal to D; therefore
BALM is equal to C, and the angle ABM equal to D.C 15.
Wherefore, &c.
b 29.
C
ز
ܪ
C 17, Cord
e
PRO P. XLV. PROB.
To
o make a parallelogram equal to a given right lined figure,
having arz angle in it equal to a giveir right lined angle.
24
Τ Η Ε Ε L Ε Μ Ε Ν Τ S
a 42
b
44.
C 220
Book I. It is required to make a parallolegram equal to a given right
lined figure ABCD, having an angle in it equal to the right
lined angle E : Join DB, and make the parallelogram FH e-
qual to the triangle ABD“; the angle FKH equal to E. Up-
on the right line GH make GM equal to DCB, and the angle
GHM equal to Eb; then FM is the parallelogram equal to
ABCD.
For, becauſe FH is a parallelogram, the angles FKH, GHK,
are equal to two right angles"; but the angles GHM, FKH,
are each equal to the angle E ; therefore equal to one another.
Add GHK to both, then the angles GHM, GHK, are equal
to FKH, GHK, that is, equal to two right angles; therefore
KHM is a right lined. For the ſame reaſon FGL is a right
lined; but FK, LM, are each parallel to GH; therefore pa-
e 30. and rallel to one another. Wherefore FM is a parallelogram e-
qual to the right lined figure ABCD, and an angle FK M equal
to E. Wherefore, &c.
Cor. Hence a parallelogram may be made equal to a given
right lined figure of any number of ſides ; for a parallelogram
can be made equal to any triangle upon any given right line.
d 146
e
conſtruct.
ܪ
PRO P. XLVI. PRO B.
To a .
O deſcribe a ſquare upon a given right line.
II.
ს 3•
C 31.
It is required to deſcribe a ſquare upon the given right line
AB.
From the point A, in the given right line AB, draw the per-
pendicular AC? ; cut off AD equal to AB b; through D draw
DE parallel to AB“, and BE parallel to AD, then ÅDEB is a
parallelogram, the oppoſite ſides of which are equal d ; that is,
e by conſtr. DE equal to AB, and BE to AD; but AD is equal to AB e;
therefore the four ſides are equal to one another. But the
angles ADE, BAD, are equal to two right anglese ; and BAD
is a right angle; therefore ADE is likewiſe a right angle ;
but the oppoſite angles of every parallelogram are equald
& Def. 30. Therefore ADEB is a ſquare. Wherefore, &c.
d 34
е
و
f 29.
2
P = 0 P, XLVII. T H F 0 P.
IN
Nevery right angled triangle the Square deſcribed upon the fide
fubtending the righi angle is equal to the ſquares of the ſides con-
taining the right angle.
Let
*
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OT EUCLI U.
23
a
.ر، و
a
Let ABC be a right angled triangle, the ſquare of the fide
BOOK I.
BC ſubtending the right angle is equal to the ſquares of the
fides BA, AC, containing the right angle. Upon BC deſcribe
the ſquare BDEC *; upon BA, AC, the fquares LG, AK; a .16.
a
through A draw AL parallel to BD, or EC 5; juin AD, FC, b 31.
BK, AE.
Then, becauſe BAC, BAG, are each right angles", GAC is
a righiline. For the ſame reaſon BAH is a right line; like- c !.!.
wiſe the angles DBC, ABF, are right anglesa; adı ABC to
both, then the whole angle FBC is equal to ADD); and AB, I dixo 2.
BD, are equal to FB, BC; and the angle FBC 10 ABD ;
therefore the triangles ABD, FBC, are equal; but the paral- e ..
lelogram BL is double the triangic ABDf; anů BG is double f 41.
TBC, or ABD; therefore the parallelograms GB, BL, are
For the ſame reaſon LC is equal to CH; but BL, 8 Ax. 6.
LC, are equal to the ſquare of BC; therefore the ſquares of
BA, AC, are equal to the ſquare of BC. Wherefore, &c.
equal
PRO P. XLVIII. THE O R.
.
F the ſquare deſcribed upon one of the fides of a triangle be e-
qual to the ſquares of the other two ſides, the angle coitaineil
by theſe two ſides is a righi angle.
If
Let the ſquare of the fide BC of the triangle ABC be equal
to the ſquares of the ſides BA, AC, the angle BAC is a right
angle.
For, let AD be drawn from the point A, at right anglesa, to a it.
AC, and equal to AB; join DC. Then, becaule the angle
DAC is a right one, the ſquare of DC is equal to the ſquares
of DA, ACB. But DA is equal to AB, and AC is common; b 47.
therefore ihe ſquares of D.1, AC, are equal to the ſquares of
BA, AC; but the ſquare of BC is equal to the ſquares of BA,
AC“, or of DA, AC: Therefore the ſquare of BC is equal to c Conſt.
the ſquare of DCd; therefore BC is equal to DC ; but BA is d Ax, 1.
equal to AD, and AC common; therefore BA, AC, are equal
to DA, AC, and the bales BC, DC, equal; therefore the
angle BAC is equal to the angle DAC But DAC is a right e 8.
angle; therefore BAAC is a right angle. Wherefore, &c.
a
D
Τ Η Ε
THE
Ε L Ε Μ Ε Ν Τ S
E L E M N S
OF
1
E U C
U
L
I D.
Β Ο Ο Κ ΙΙ.
DEFINITIONS.
UE
1.
BOOK II. VERY right angled parallelogram is ſaid to be contains
ed by two right lines containing the right angle.
II.
In every parallelogram, either of the two parallelograms that
are about the diameter, together with the complement, is called
a gnomon.
PRO P. I. T H E O R.
IF
F there be two right lines, and one of them divided into any
number of parts, the rectangle contained by the whole, and di-
vided line, is equal to all the rectangles contained by the whole
line, and the ſeveral parts of the divided linc.
a II. I
IlI.
b
Let A, BC, be the two right lines, one of which, viz. BC,
is divided into any number of parts, as D, E; the rectangle
contained by A, BC, is equal to the rectangles contained by A,
BD; A, DE; A, EC.
For, from the point B draw BF, at right angles, to BC^;
make BG equal to Ab; through G draw GH parallel to BC;
and through the points D, E, C, draw DK, EL, CH, each pa-
rallel to BGC.
The rectangle BK is that contained by BD, BG; for BGd is
equal to A; the rectangle DL is contained by A, DE; and
EH
3. I.
C 31. I.
a Const.
OF EUCLID.
27
;
EH by A, EC ; for DK, EL, CH, are each equal to BG, that Book II.
is, equal to A; but the rectangle BH is equal to the rectangles
BK, DL, EH; and BH is contained by A, BC; therefore the e 3t. 1.
rectangle by A, and BC, is equal to the rectangles by A,
BD; A, DE, and A, EC. Wherefore, &c.
PRO P. II. THE O R.
IF
a right line be any how cut, the rectangles contained by
the whole line, and each of the ſegments, are equal to the Square
of the whole line.
a 40. 1.
Let the right line AB be any low cut in C, the rectangles
contained by AB, BC, and AB, AC, together, are equal to the
ſquare of AB. Upon AB deſcribe the ſquare ADEB a;
a; thro'
C draw CF parallel to ADb, or BE; then, becauſe AD is e-b 31. a.
qual to ABC, the rectangle under AD, AC, is equal to the rec-
tangle under AB, AC, and the rectangle under EB, BC, is e-
qual to the rectangle under AB, BC; but the rectangle under
AD, AC, that is, the rectangle AF, together with the rec-
tangle under EB, BC, that is, CE, are equal to the ſquare of
AB; that is, the ſquare AE. Wherefore, &c.
>
PRO'P. III. THE O R.
OR
Fa right line be any how cut, the rectangle under the whole
line, and one of the parts, is equal to the rectangle under the
two parts, together with the square of the firſt mentioned part.
9
Let the right line AB be any how cut in C, the rectangle
under AB, BC, is equal to the rectangle under AC, CB, to-
gether with the ſquare of BC. Upon BC deſcribe the ſquare
BCDE a; produce ED to F, and draw AF parallel to CD or
BE; the rectangle under AB, BC, that is, AE, is equal to the
rectangles AC, CD; that is, the rectangle under AC, CB, and
the ſquare of CB. Wherefore, &c.
a
a 46. I.
PRO P. IV. THE O R.
IT
Fa right line be any how cut, the ſquare of the whole line is e-
qual to the ſquares of the two parts, with twice the rectangle
under theſe parts.
Let
>
28
THE ELEMENTS
9
a 5. I.
d
e
Воок ІІ.
Let the right line AB be any how cut in C, the ſquare of
AB is equal to the ſquarcs of AC, CB, and twice the rectangle
under AC, CB.
For, upon AB defcribe the ſquare ADEB; through C
draw CT parallel to AD, or BE; draw DB, cutting CF in G;
through which point draw HGK paralld to AB, or DE. (Then
the figure is faid to be conſtructed.) Now, becauſe AB is equal
to AD, the angle ADB is equal 10 ABD"; and the angle CGB
to ADB"; therefore the angle CDG is equal to CGBC; there-
C Am. 1. 1. forc CG is equal to CB; therefore CGKB is equilateral €;
but the angles BCG, CBK, are equal to two right angles, and
CEK is a right anglef; therefore BCG is likewiſe a night
angle; therefore all the angles are right ones; and CGK) is
& Def. 30, a iquare 8. For the ſame reaſon HF is a ſquare. But the rec-
tangles AG, GE, are equalb; and AG is the rectangle under
11 43. I.
AC, CB; for CG is equal to CE; but the ſquares HF, CK,
with the recargles AG, GE, make up the ſquare of AB: There-
fore the ſquare of AB is equal to the ſquares of AC, CB, and
twice the rectangle under AC, CB. Wherefore, &c.
Cor. Hence every parallelogram about the diameter of a
fquare is a ſquare.
1) 29 I.
с Ax.
d 6. I.
€ 3+ I.
f 46.1.
b
•
୬
C
.
1.
PRO P. V. THE O R.
9
IF
a right line be cut into two cuial parts, and into two unequal
perts, the riangle under the two unequiil parts, together
with the jaziare of the intermediate part, are equal io the jquare
of half the line.
Let the right line AB be cut equally in C, and unequally in
D, the rectangle under AD, DB, tugether with the ſquare of
CD, recqual to the ſquare of CB.
For, upon CB deſcribe the fiziare CETB; conſtruct the fi-
gure, and produce OL 10 K; and through A draw AK parallel
to
;
7
to IC.
a.
a 30. I.
43. I.
C. ix. 2,1
d Ax. l. I.
I
Cor, of
The parailelogranis AL, CO, are' -'qual", ond CH is equal
to HFb; add Do in lioth; thun CO is equal10 DF'; therefore
coi
AL is likewiſe equal to DH '; ad CH to both; then the rec-
tangle AH is equal to the gnomo. LDF; aud LG, that is, the
ſquare of CD, to loth ; tb.n the rectarigle AB, tisat is, the
thin A5
rectangle under AD, DB, with LG, thai is, the iquare of CD,
are equal to the gnomon IDF, and LG; that is, equal the
squarcof CB. Wherefore, &c.
e
ģ
PROP
OF EU C L I D.
29
Book II.
PRO P. VI. T H E O R.
I
Fa right line be divided into equal parts, and another line add-
I cd to it, the refiangle contained by the whole and added linc
as one ſide of the rectangle, and the added line for the other fidi,
together with the ſquare of half the line, are equal to the ſquare of
the half and added line, as one ſide of the ſquare.
Let the right line AB be biſected in C, and BD added to it,
the rectangle under AD, DB, together with the ſquare of BC,
are equal to the ſquare of CD.
Deicribe the ſquare (EFD; conſtruct the figure, and com-
pleat the parallelogram under AC, CL.
Then the parallelograms AL, CH, are equal"; but CH is e-a 36. 1.
qual to HF b; add BM to both; then CM is equal to BF; add b 43. I.
AL to both; then AM is cqual to the gnomon CMG. To
each add LG, that is, the ſquare of CB "; then AM, LG, arec Cor. 4.
equal to the gnomon CMG, and LG; that is, the rectangle un-
der AD, DB, for DM is equal to DB, together with the
ſquare CB, are equal to the ſquare of CD. Wherefore, &c.
PRO P. VII.
THEOR.
F a right line be any how cut, the ſquare of the whole line, and
one of the parts, is cqual to twice the rectangle contained
by the whole line, and ſaid part, together with the ſquare of the e-
ther part.
a
>
b
و
Let the right line AB be any how cut in C, the ſquares of
AB, BC, are equal to twice the rectangle under AB, BC, and
the ſquare of AC.
Upon AB deſcribe the ſquare ADEB", and conſtruct the fi- a 45. I.
a
gure; then, becauſe the rectangle AF is equal to CEV, and AF,
643. I. and
CE, together, are equal to twice AF, that is, equal to the gno-
mon AFK, together with the ſquare of CB, that is, CF'; add c Cor. 4,
HK to both; then twice AF, and HK, are equal to the gnomon
AFK, and the ſquares of AC, BC; that is, to the ſquares of
AB, BC. Wherefore, &c.
Ax.2.1
C.
.
7
PROPE
30
THE E L E M E N T S
Book II,
PRO P. VIII. THEOR.
1
TE
a right line be cut into two parts, four times the rectangle
under the whole line, and one of the parts, together with the
Square of the other part, are equal to the ſquare of the whole line,
and the firſt part taken as the ſide of the ſquare.
b
a Cor. 4.
b36, 1.
с
€ 29. 1.
d
34. J. and
Let the right line AB be any how cut in C, four times the
rectangle under AB, BC, together with the ſquare of AC, are
equal to the ſquare of AD; that is, AB produced to D, ſo that
BD equal BC. Upon AD deſcribe the ſquare AEFD, and
conſtruct the double figure. Then, becauſe BN, GR, are
ſquaresa, and CK, BN, are equal parallelograms b; but the ſides
CB, BK, are equal, and CBK is a right angle, for it is equal
to BDN°; therefore CK is a ſquared. For the ſame reaſon, KO
def. 30.1. is a ſquare; therefore CK, BN, GR, KO, are each ſquares;
but they are conſtitute upon equal right lines ; therefore equal
to one another, and, together, quadruple KC. But the rec-
tangle AG is equal to MP, and PL to RF; but MP is equal
43, 1.
to PLf; therefore the four rectangles are quadruple AG;
and the four ſquares and four rectangles quadruple the rec-
tangle AK, that is, the rectangle under AB, BC; add the
ſquare XH, that is, the ſquare of AC; then four times AK, that
is, four times the rectangle under AB, BC, together with the
ſquare of AC, are equal to the ſquare of AD. Wherefore,
&c.
PRO P. IX.
THE OR.
IFS
a right line be cut into two equal parts, and into two uncqual
parts, the ſquares of the two unequal parts are double the ſquare
of the half line, and double the ſquare of the intermediate part.
Let the right line AB be cut equally in C, and unequally in
D, the ſquares of AD, BD, are double the ſquares of AC,
CD.
For, through C draw CE, at right angles, to AB, and equal
to AC, or CB ; join EA, EB; through D draw DF parallel to
3
to CE, and FG through F, parallel to AB ; join AF.
Then, becauſe AC is equal to CE, and the angle ACE a
Cor. 32. right angle, the angles AEC, EAC are each half right angles a,
and the ſquares of AC, CE double the ſquare of AC; but the
b 47. 1:
{quare of AE is equal to the ſquares of AC, CE 6; therefore,
double
2
e .
a
b
OF EUCLI D.
31
с
a
0
.
double the ſquare of AC. For the ſame reaſon, the angles CEB, Book II.
EBC are each half right angles ; but the angle EGF is a right cor
angle; therefore GFE is half a right angle a ; therefore the ſides c 29. 1.
EG, GF are equal d; but the ſquare of EF is equal to the ſquares d 6. 1.
of LG, GF, or double the ſquare of GF or CD '; but the ſquares e 34. 1.
of AE, EF are equal to the ſquare of AF, for the angle AEF is
a right angle ; but the ſquares of AE, EF are double the ſquares
of AC, CD; therefore the ſquare of AF is double the ſquares
of AC, CD; but the angle DFB is half a right angle ; for it is en
qual to CEB °; therefore, DFB, DBF are each half right angles;
therefore FD, DB are equal; but the ſquare of AF is equal to
the ſquares of AD, Drb, or DB; therefore, the ſquares of AD,
DB are double the ſquares of AC, CD. Wherefore, &c.
و
.
PRO P. X. THE OR.
T .
IF
F a right line be cut into two equal parts, and another right
line added to it, the ſquare of the whole and added line
taken as one line, and the ſquare of the added line, are double the
Square of the half line, and double the ſquare of the half and added
line, taken as one line.
ܟ
Let the right line AB be biſected in C, and BD added to it,
the ſquares of AD, DB are double the ſquares of AC,
CD.
For, from the point C, draw CE perpendicular to AB, and
equal to AC or CB ; join AE, EB ; through E, draw EF pa-
rallel to AD, and through D, draw DF parallel to CE.
Becauſe AC is equal to CE, and the angle ACE a rigat an-
gle, each of the angles AEC, EAC are half right angles; and
therefore the ſquare of AE is equal to the ſquares of AC, CE,
or double the ſquare of ACX. For the ſame reaſon, CEB, CBE a 47. I.
are each half right angles; therefore AEB is a right angle; but
the angles FEC, ECD are equal to two right angles b, and ECD b 29. 19
is a right angle; therefore CEF is likewile a right angle; there-
fore CF is a rectangle"; therefore, the angles DTE, FEC are equal
to two right angles b; therefore, DEF, TEB are leſs than two right
angles d; therefore, FD, EB will meet one another, which let be d. Cor. 17.
in G. But CEF is a rightangle, and CEB half a rightangle; there- 1.
fore, FEB is half a right angle, and IGFislikewilè half a right an-
gle; therefore, EF is equal to FG, and the ſquare of EG equal e 6, 1.
to the ſquares of EF, FG “, or double the ſquare of EF, or CD;
therefore, the ſquares of AE, EG are double the ſquares of AC,
CD; but the ſquare of AG is equal to the ſquares of AE, EG,
and
b
с
e
c Def, I.
9
a
2
a
32
Τ Η Ε Ε L Ε Μ Ε Ν Τ S
Book II' and likewiſe equal to the ſquares of AD, DG ; for the angles
AEG, and ADG are each right ones; therefore, the fquares of
AD, DG, or DB its equal, are double the ſquares of AD, CD.
Wherefore, &c.
I
PRO P. XI. PRO B.
O cut a given right line ſo, that the rectangle contained un-
der the whole line, and one of the parts, be equal to the
Square of the other part.
T'
a 46. I.
9
b 6
و
c Conſt.
Upon any given right line, as AB, deſcribe the ſquare
ABDC ;
a; biſect AC in E;join EB, and produce EA to F; make
EF equal to EB ; upon AF, deſcribe the ſquare FGHA, and
produce GH to K; then AB is ſo cut in the point H, that the
rectangle under AB, BH is equal to the ſquare of AH.
For the rectangle under CF, FA, together with the ſquare of
AE, is equal to the ſquare of EF b; but EF is equal to EB “, and
the ſquare of EB is equal to the ſquares of BA, AE d; therefore,
the rectangle under CF, FA, together with the ſquare of AE,
are equal to the ſquares of BA, A E. Take the ſquare of AE from
both, there remains the rectangle under CF, TA, that is, the
rectangle under CF, FG, that is, FK, equal to the ſquare of
AD. Take AK from both, there remains FH equal to HD; but
FH is the ſquare of AH, and HD the rectangle under AB,
BH, for BD is equal to AB. Wherefore, &c.
و
d 47. I.
PRO P. XII. THE OR.
TO
Nevery obtufe angled triangle, the ſquare of the ſide ſubtending
the obtufe angle, is greater than the Squares of the ſides con-
taining the obtuſe angle, by twice a rzetangle under one of the
fides containing the obtufe angle, and that part of the ſide produ-
ced, lying betwixt the obtuſe angle, and perpendicular let fall from
the oppoſite angle.
a 12. I,
Let BAC be the obtuſe angle of the triangle ABC; pro.
duce the fide CA till it meet the peopendicular BD, let fall
from the point B2. The ſquare of BC is greater than the
ſquares of BA, AC, by twice the rectangle under CA,
AD.
For the ſquare of BC is equal to the ſquares of BD, DC6;
but the ſquare of DC is equal to the ſquares of DA, AC, and
twice
}
47. I.
1
OF EUCLID,
33
twice the rectangle under AD, AC°; but the ſquare of AB Book II.
is equal to the ſquares of BD, DA; therefore the ſquare of
.
BC is equal to the ſquares of BA, AC, and twice the rectangle 4.
under DA, AC; therefore the ſquare of BC is greater than
the ſquares of BA, AC, by twice the rectangle under DA, AC.
Wherefore, &c.
C4
b 77. 1.
PRO P. XIII. THE O R.
I Nevery acute angled triangle, the ſquare of the fide fubtending
the acute angle, is leſs than the ſquares of the fide containing
the acute angle, by twice a rectangle contained under one of the
fides about the acute angle, and that part of the ſide lying between
the acute angle and the perpendicular let fall from the oppoſite
angle.
a
9
b
Ն
b
Let B be an acute angle in the triangle ABC ; from the
angle A let fall the perpendicular AD, cutting BC in D ; a 12. 1.
the ſquare of AC is leſs than the ſquares of AB, BC, by twice
the rectangle under CB, BD.
For the ſquare of AC is equal to the ſquares of AD, DCD; 47. I.
and the ſquare of AB is equal to the ſquares of AD, DB b;
but the ſquares of BC, BD, are equal to twice the rectangle
under BC, BD, together with the ſquare of DC“; therefore the c 7.
ſquares of AB, BC, are equal to the ſquares of AD, DC, and
twice the rectangle under CB, BD; but the ſquare of AC is e-
qual to the ſquares of AD, DC; therefore the ſquare of AC is
leſs than the ſquares of AB, BC, by twice the rectangle under
CB, BD. Therefore, &c.
PRO P. XIV. PROB.
T
O make a ſqucre equal to a given right lined figurc.
ز
و
Make the rectangle BCDE equal to a given right lined
figure A"; If BE be equal to LD, then BCDE is a ſquare ; a 45. I.
and what was required is done. If not, produce EE to F;
make EF equal to ED, and biſect BF in Gb; with the center b 10. 1.
G, and diſtance GB, deſcribe a ſemicirclu BHF; produce DE
to H, and join GH.
Then
E
a
1
34
1
THE ELEMENTS
e
CS.
BOOK II. Then the rectangle under BE, EF, together with the
ww'ſquare of GE, are equal to the ſquare of GFC, or GH d; but
the ſquare of GH is equal to the ſquares of GE, EH;
d Def. 15 therefore the rectangle under BE, EF, together with the
ſquare of GE, are equal to the ſquares of HE, EG. Take
the ſquare of GE from both, and the rectangle under BE,
EF, that is, BD, is equal to the ſquare of EH. Where- ,
fore, &c.
1.
© 47. I,
!
3
THE
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1
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B.
D E
C: Al
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Prop. 12.
F
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A
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Τ Η Ε
Ε L Ε Μ Ε Ν Τ S
OF
E U
C L L 1 D.
I
BOOK
III.
DEFINITIONS.
EQUA
QUAL circles are ſuch whoſe diameters are equal.
1.
.
Book III.
II.
A right line is ſaid to touch a circle, when drawn to the ſame,
and being produced, does not cut the circle.
III.
Circles are ſaid to touch each other, which, meeting, do not cut
one another.
IV.
Right lines in a circle are ſaid to be equally diſtant from the
center, when perpendiculars drawn from the center to each
of them are equal, and that line upon which the greateſt
perpendicular falls is the leaſt line.
V.
Definition 19th, 1.
VI.
An angle of a feginent is the angle contained by the right line
and circumference of the circle.
VII.
An angle is ſaid to be in a ſegment, when right lines are drawn
from ſome point in the circumference to the ends of that
line which is the baſe of the ſegment, which lines contain the
angle.
VIII. But
36
THE ELEMENTS
1
Book III.
VIII.
nu But, when the right lines containing the angle do receive any
part of the circumference, then the angle is ſaid to ſtand upon
that circuinference,
IX.
A fictor of a circle is that figure which is contained by two
right lines, drawn from the center, and the circumference be-
tween theni.
1
Similar ſegments of circles are thoſe which include equal angles
or whcreof the angles in them are equal.
PRO P. I. PROB.
1
T TA
o find the centre of a circle.
a IO. I.
ს II. I.
Required to find the centre of the giren circle ABC.
Draw in it the line AB, which bilea in D, by the right line
CD at right angles to AB), and produce CD to E ; biſect EC
in F; which point is the centre of the circle ABC.
If not, let G be the centre; join GB, GA, GD; then, be-
cauſe AD is equal to DB, DG is common, the baſe AG is e-
c deſ. 15. 1. qual to GB“, and the angle ADG to GDBd; therefore, each of
them is a right angle; but I'DB is a right angle; therefore,
GDB is equal to FDB, a part to the whole, which is impor.
e Ax. 9. 1. fiblee; therefore, no point but F can be the centre. Where-
fore, &c.
Cor. Hence, if, in a circle, any right line cut another right
line into two equal parts, the centre of the circle will be in that
line which cuts the other into two equal parts.
ds. ;.
PRO P. II. THE O R.
a
TF any two points be al umed in the circumference of a circle, the
right line joining theſe points will fall within the circle.
a I.
Let the circle be ABC, A and B the points in its circumfe-
rence, the right line AB, joining theſe points, will fall within
the circle. Find D the centre of the circle ; join DA, DB,
and draw DF, cutting the right line AB in the point L; then
b def. 15. 1. the right lines DA, DE, DF, are equalb. But DF is greater
¢ Ax. 9. 1. than DE®; therefore, DA, DB, are likewiſe greater than DE;
but DB, DA, reach the circumference; therefore DE does not
reach
OF EUCLI D.
37
reach the circumference; therefore the right line AB is within Book III.
the circle. Wherefore, &c.
Cor. Hence, if a right line touches a circle, it will touch it
only in one point.
P R O P. III.
T H E OR:
>
IF, in a circle, a right line be drawn through the centre, cutting
another line not drawn through the centre, into two equal
parts, it fall cut it at right angles ; and if it cut it at right
anglis, it ball cut it into two equal parts.
a
2
Let ABC be the given circle, CD the line paſſing through
the centre, cutting the right line AB, not paſſing through the
centre, into two equal parts, it will cut it at right angles; and
if the angles AFE, EFB, be right anglos, AF will be equal to
FB.
For, find the centre Eạ; join EA, EB; for, becauſe AF is e- a so
qual to FB, and FE is common, the two fides AF, FE, are equal
to BF, FE, and the baſe A equal to EBO; therefore, the angle b def. 15.1.
AFE is equal to EFB', and each is a right angled; therefore, à Nef. 1e.
CD cuts the right line AB at right angles. If EFA, EFB, are
right angles, AF is equal to FB; for, becauſe AE is equal to
EB, and EF common, the two ſides AE, EF, are equal to BE,
e Hyp.
EF, and AFE, EFB, right angles®, and the angle EAB equal to f 5.1.
EBAF; therefore, the remaining angle AEF is equal to FEB8; & Cor. 33.
therefore the baſe AF is equal to FBh. Wherefore, &c.
C 8. 1.
I.
I.
32.
I.
h 4. I.
PRO P. IV. TH E O R.
OR
IF two right lines are drawn in a circle, neither of then paſſing
through the centre, they will not mutually biſečt each other.
و
Let two right lines AC, BD, not paſſing through the center,
be drawn in the circle ABCD, cutting each other in the point
£, they will not mutually biſect each other ; for, if poſſible, let
AE be equal to EC, and BE to ED; find the center F, and join
TE; then, becauſe FE, paſſing thro' the center, cuts the right
Jine AC, not paſſing through the center, into equal parts, the
angics FEA, FEC, are right anglesa ; and, becauſe FE biſects BD, a 3.
FEB is a right anglea; therefore, FEA is equal to FEB, a part
to the whole, which is impoflible; therefore, AC, BD, do not
mutually biſect each other. Wherefore, &c.
a
9
PROP
38
THE ELEMENTS
1
Book III.
PRO P. V. THE O R.
IF
If two circles cut each other, they cannot have the ſame
center.
Let the two circles ABC, CDG, cut each other, they cannot
have the ſame center.
if poſſible, let E be the center of both, draw CE to the point
of fedtion C, and EFG through any other point; then, becauſe
E is the center of the circle ABC, EC is equal to EF ; and, be-
.
caufe L is the center of the circle CDG, ÉC is equal to EG;
h. Ax. 11. therefore, EF is equal to EG), a part to the whole, which is
impoſſible; wherefore E is not the center of both. Wherefore, &c.
a
a Def. 15.
PRO P. VI.
Ꮲ
VI. THE O R.
O
IF F two circles touch each other inwardly, they have not the
fame center.
Let the two circles ABC, CDE, touch each other inwardly in
the point C, they have not the ſame center. If poſſible, let it
be F; join FC, and draw FB through any other point.
Then, becauſe F is the center of the circle ABC, CF is equal
3 def. 15.1. to FB; for the ſame reaſon CF is equal to FE; therefore FE
is equal to FB, that is, a part to the whole, which is impoſſible.
Wherefore, &c.
b 37. I.
PRO P. VII.
THE O R.
IF some point is taken in the diameter of a circle, which is not
the center, from that point if ſeveral right lines are drawn to
the circumference, the greateſt of theſe right lines is that part of
the diameter in which the center is, and the remainder of the dia-
meter is the leaſt ; of the other lines, the neareſt to that palling
through the center is greater than that more remote ; and, on each
ſide of the diameter, only two right lines can be drawn from that
point to the circumference equal
to one another.
a
Let F be a point in the diameter of the circle ABCD, which
is not the center, and from it be drawn FA thro’ the center, and
FB, FC, FG, any how to the circumference, FA is the great-
eſt line, FA is greater than FB, FB greater than FC, FC great-
Sen
OF EUCLI D.
39
3
2
er than FG, and FD the leaſt. And from the point F only two Book III.
right lines can be drawn equal to one another on each ſide of
diameter; for, find the center E; join BE, CE, and GE.
Then, becauſe E is the center of the circle, EA is equal to EB;
add EF to both; then AF is equal to BE, EF; but BE, EF, are
greater than BF4; therefore Al is greater than BF; but BE is a 20. I.
equal to CE, and EF common; therefore BE, EF, are equal to
CE, EF; but the angle BEF is greater than CEFb; therefore, the li Ax, S.
baſe BF is greater than CFC. For the ſame reaſon CF is greater c 24. s.
than GF; likewiſe the two fides GF, FE, of the triangle GEF,
are greater than GE, that is, than ED; take EF from both,
there remains GF greater than FDd; therefore, AF is the great- d Ax, 5. I.
eſt right line, and FD the leaſt.
Laſtly, on each fide of the diameter, from the point F, only
two right lines can be drawn equal to one another; for, at the
point E, with the right line EF, make the angle FEH equal
to FEG, then the bale FH is cqual to GF. If any other right e 4. 1.
line can he equal to FG, let it be FK, that is, a line nearer to
that paſſing through the center, equal to one more remote,
which cannot be. Wherefore, &c,
ܙ
PRO P. VIII. TH E O R.
9
Fa point be taken without the circle, and from it right lines be
drawn, one of which paling through the center, and the other
falling upon the concave part of the circumference, the greateſt of
theſe lines is that paling through the center; and the line nearer
to that, paſſing through the cenier, is greater than that more re-
wote; of thoſe falling upon the convex part of the circumference,
that which lies betwixt the point and the diameter is the lui!i lini,
and that line nearer to that paſſing through the center, is ie's than
that more remote ; and, on each ſide of the diimcters oily two lines
can be drawn from that point, falling either on the COP!C17VC Or Coil-
pex part of the circumference equal to one another,
miot
ر
Let any pcint D be taken without the circle ABC; draw DA,
DE, DF, DC, to the concave part of the circumference; of thefo
lines DA, which paffes through the center, is the grenteſt; DE
is greater than DF, and DF than DC. Of theſe that fall upon
the convex part of the circunference, DG is the leaſt, DK is
leſs than DL, and DL than DH; on each ſide of DG only two
right lines can be drawn equal to each other, either on the con-
vex or concave part of the circumference. For, find the center
M; draw ME, MF, MC, MH, ML, MK. Now, becauſe MA
is
>
40
THE ELEMENTS
a 20. 1.
b 24. I,
I.
d Ax, 5. I.
.
e 21. I.
с
Book III. is equal to ME, add MD, which is common to both, then AD
Vis equal to DM, ME; but DM, ME, are greater than DE-; there-
fore DA is greater than DE; but DM, ME, are equal to DM;
MF, and the angle DME greater than DMF; therefore DE is
greater than DFO. For the fainc reaſon, DF is greater than
DC; wherefore DA is the greateſt of the right lines. falling on
the concave part of the circumference. Again, becauſe DK
c Def
. 15, KM are greater than DMa, take the equal lines KM, GM“, from
both, there remains DG leſs than DK4; but K is a point taken
within the triangle DLM; therefore DK, KM, are leſs than
DL, LM; take MK, MLC from both, there remains DK leſs
than DL. For the ſame reaſon, Lis leſs than DH; wherefore
DG is the leaſt line, and DK leſs than DL, &c.
Likewiſe, from the point D on each ſide of the leaſt line,
only two right lines can be equal to each other, falling on the
convex part of the circumference. For, make the angle DMB
equal to the angle DMK, and join DB ; then, becauſe DM, MB,
are equal to DM, MK, and the angle DMB to DMK, the bale DB
f 4. I.
is equal to DKF. If any other right line can be equal to DB,
let it be DN; that, is a line nearer to the leaſt line equal to one
more remote, which cannot be. Neither can more than two
equal right lines fall upon the concave part of the circunference
on each ſide of the diameter from the fame point.
For, let the angle AMO be made equal to AME, join MO,
$ 13. 1, DO; then theangles AMO, DMO, are equal totwo right angless,
and AME, DME, likewiſe equal to two right angles; but AMO
h Ax. 3. 1. is equal to AME; therefore, the angle DMO is equal to DME";
but DM, NE, are equal to DM, MO, and the angle DME to
DMO; therefore, the baſe DO is equal to DEI. If any other
right line can be equal to DO, let it be DP, that is, one nearer
to that paſſing through the center, equal to one more remote.
Wherefore, &c.
ز
i 4. In
PRO P. IX. THEOR.
IF
F a point be aſſumed in a circle, and from it be drawn more than
two right lines to the circumference equal to one another, that
point is the center of the circle.
Let the point D be affumed in the circle ABC, and from it
be drawn, to the circumference, the right lines DB, DC, DA,
equal to one another, D is the center of the circle. If not, let
E be the center, join D, E, which produce to F and G; then is
T
FG the diameter, and D is ſome point in it, not the center;
therefore
>
i
OF EUCLI D.
41
therefore DG is greater than DC, DC than DB, and DB Book III
than DA"; but DC, DB, DA, are equal b; and likewiſe not e-
qual; which is impoſſible; therefore no point but D is the cen- a 7.
ter of the circle. Wherefore, &c.
b Hyp.
PRO P. X. THE O R.
O
0
N E circle cannot cut another in more than two points.
For, if poſſible, let the circle ABC cut the circle DEF in the
points B, G, F; let K be the center of the circle ABC; join
BK, KG, KT. Now, becauſe K is a point within the circle
DET, from which there is drawn to the circumference the right
lines BK, KG, KF, equal to one another; therefore K is the
center of both circles a; which is impoſſible b.
. Wherefore, ;
&c.
a 9.
b 5
PRO P. XI. T H E O R.
I
F two circles touch one another inwardly, a line joining their
centers will fall on the point of contact.
ܪ
>
Let the two circles ABC, ADE, touch each other inwardly
in the point A ; let F and G be the centers of the circles ABC,
ADE, then the line joining the centers F, G, will paſs through
the point A. If not, let the right line joining the centers F, G,
cut the circles in the points D, H; join GA; then, becauſe Fis
the center of the circle ABC, FA is equal to FH 2. For the a def. 15. 1.
ſame reaſon, GD is equal to GA; but GA, GF, are greater
than AFb; therefore DF is greater than AF; therefore greater b 20. 1.
than HF; and likewiſe leſs"; which is impoffible; therefore the c Ax. 9. J•
;
line joining the centers will not paſs through any other point
than A. Wherefore, &c.
PRO P. XII. THEO R.
I
F two circles touch one another outwardly, a line joining their
centers will paſs through the point of contact.
Let the two circles ABC, ADE, touch one another outwardly
in the point A; a right line, joining their centers, will paſs
F
through
a
42
1
THE ELEMENTS
Book IIT. through the point A. If not, let F, G, be the centers of
the two circles, and the right line FG joining them; cut the
circles in C, D ; join FA, AG. Becauſe F is the center of
3 def. 15.1. the circle ABC, FA is equal to FC a; and, becauſe G is the cen-
ter of the circle ADE, GA is equal to GD"; add DC; then
the whole FG is greater than FA, AG; and likewiſe leſs b;
which is impoſſible. Wherefore, &c.
a
b 20. ,I
PRO P. XIII. T H E O R.
Τ Η Ε Ο
O
NE circle cannot touch another, either outwardly or inwardly,
in more than one point.
0
a II,
The circles ABC, DFD, cannot touch one another inwardly in
more than one point : for, if poſſible, Ict them touch in B, Ꭰ ;
;
Jet G be the center of the one circle, and H of the other ;
then
BG is equal to GD, and greatcr than HD; therefore BH is
much greater than HD; but H is the center of the circle BDF a;
therefore BH is equal to HD; and likewiſe greater; which is
impoſlible: Therefore the circles ABC, BPD), cannot touch
one another inwardly in the points B, D. Let the circle AKC
touch the circle ABC outwardly in the points A, C, if poſſible;
join AC; then is 2 within the one circle, and without the o-
ther; which is impoſible. Wherefore two circles, &c.
PRO P. XIV. THE O R.
Nr number of equal right lines, drawn in a circle, are e-
qually diſtant from the center ; and, if they are equally
diſtant jroni tke center, they are equal to one another.
A
a
I
a 31
b 7. I.
1
Let AB, CD, be two equal right lines, drawn in the circle
ABD; and E, the center of the circle, and from it let fail the
perpendiculars ! F, EG, they will be equal to one another; for,
join TA, EC ; then are the rightlit.SAB, CD, buitcted by the
right lines EF, EGạ; the ſquare of AE is equal in the quares
of AF, HE B; and the ſquare of 1.C equal to the ſquares cË CG,
CU; therefore the iqures of AF, FE, are equal to the quares
OF CG, GPS, but the ſquare of Al is equal to ile fquare of
15,6" the fo a the Square of 13 is equal to the ſquare of EG
kinjvejre A:.,:'), are equally diſtant from the center.
y
1/3?ticgu. ] to BG, th: 12 B will be equal to CD;
003 O AF, 13, are equal to the squares of CG
C
}
I,
1
5.
9
9
$
1
OF EUCLI D:
43
s
į
er
GE; but the ſquare of EF is equal to the ſquare of EG ; there. Book III.
fore the ſquare of AF is equal to the ſquare of CG“; but
AB is double Ara, and CD double CG ; therefore AB is equal e Ax. s.
to CD. Wherefore, &c.
a 3
fax. 6.
PRO P. XV. THE O R.
OR
T.
HE diameter of a circle is the greateſt right line in ii,
and the line neareſt to the diaincter is greater than that
more remote; and on each ſide of the diameter only two right lines
can be drawn equal to one another.
a
Let ABC be a circle, whoſe diameter is AD; let MN, FG,
be drawn any how in the circle; then AD is the greateſt line;
AD greater than MN, and MN greater than FG. Find the cen-
ter E; draw EM, EN, EF, EG, then AE, ED, are equal to
ME, EN; but ME, EN, are greater than MN a; therefore ADa 20. 1,
is greater than MN; likewiſe ME, EN, are equal to FE, EG,
and the angle MEN greater than FEG; therefore MN is greater
thian FG b: So likewiſe on each ſide of AD only two right lines b 24. si
can be drawn equal to one another, viz. upon which the equal
perpendiculars fall. For, let fall a perpendicular EL upon' MN,
and draw LH equal to it, and BC at right angles to EH ; then
BC is equal to MNC. If any other right line can be equal toc 14.
MN, or BC, let it be FG, a line nearer to the diameter equal
to that more remote. Wherefore, &c.
PRO P. XVI.
THEOR.
Line drawn from the extreme point of the diameter of a
circle, at right angles to that diameter, Joall fall without
the ſame ; and between that right line and the circumference no
right line can be drawn.
.
A
i
Let ABC be a circle, whoſe diameter is AB; at the extre-
mity of which, if a right line is drawn at right angles, it ſhall
fall without the circle.
If not, let it fall within the circle, as AC; find the center,
and join CD. Then, becauſe DAC is a right angle, DCA will
be
a
44
THE ELEMENTS
a
a S. 1.
b 17. I.
.
ز
ܪ
;
a
Book III. be likewiſe à riglit angie, for DA is equal to DC“, that is, two
angles in a triangle equal to two right angles; which cannot beb;
neither can it fall upon the circle”; therefore it muſt fall with-
c Dif. 4. 1. out the circle, which let be AE; and betwixt the right line AE,
and circumference CHA, no right line can be drawn. If pof-
fible, let TA be drawn; then DAF is leis than a right angle.
From the point D, to the right line TA, a line can be drawn at
right angles to FA, falling without the circle ; which let be
DG; then, becauſe DGA is a right angle, and DAG leſs than
din. I.
a right angle, DA is greater than DG d; but DA is equal to
DH; therefore DH is greater than DG, and likewiſe leſs
which is impoffible : Therefore, betwixt the circumference
and right line AE no other right line can be drawn. Where-
fore, &c.
Cor. I. Hence the angle between the right line and cir-
cumference is the leaſt of all acute angles; and the angle be-
twixt the diameter and circumference is the greateſt acute angle
poffible.
II. Hence, likewiſe, a right line, drawn at right angles, at
the extreme point of the diameter of a circle, touches the circle
only in one point ; for, if it meet it in two points, it would fall
within the circle e.
ز
2.
PRO P. XVII. PROB.
T.
o draw a right line that will touch a given circle from a
given point without the fame.
2 I.
bil. I,
>
9
Lct BCD be the circle, and A the point without it; it is re-
quired to draw a right line from the point A, that will touch the
circle BCD. Find E the center of the circle”; join AE, cutting
the circle BCD in D. About the center E, with tlie diſtance EA,
deſcribe the circle AFG 2: the point D; araw DF at right
angļos to DE b, cutting the circle AFG in T; join IF; cutting
DRC in B, and join AB; then is AB the tangent required.
for, becauſe I is the center of both circles, the right lines
AL, EB, are equal to FE, ED, and the angle E common ;
therefore the triangle ABE is equal to FDE"; and the angle
EBA to EDF; but EDF is a right angle; therefore ABE is
likewiſe a right angle: Therefore AB is a tangent to the circle
in the point Bd, and drawn from the point A. Which was re-
quired.
с 4. І.
d cor. 16.
PRO B.
A
OF EUCLI V.
45
Book III.
PRO P. XVIII. and XIX. THE OR.
O
I
F any right line touches a circle, and from the center to the
point of contact a right line be drawn, that line will be at
righi angles to the tangent; and if, from th? fcint of contact, a
right line be drawn, palling through the circle, at right angles
io the tangent, the center of the circle will be in that line.
a
a IX. I.
с
Let ABC be a circle, and DE a right line touching it in the
point C; and if, from the center F, there be drawn a right
line FC, that line will be perpendicular to the tangent. If
not, let FG be drawn from the center F, at right angles to
DE2.
Now, becauſe FGC is a right angle, FCG will be leſs than a
right angle b; therefore FC is gre..ter than FG*; that is, FB b 17. 1.
greater than FG, a part greater than the whole ; which is impof-c 19. 1
lible. For the ſame reaſon, no right line but FC can be perpen-
dicular to DE.
2diy, If, from the point of contact C, of the tangent DE, AC
be drawn through the circle ABC, at right angles to DE, the
center of the circle will be in AC. If not, let it be in H; join
HC; then HCE is a right angled; but ACE is a right angle; d 15.
therefore HCE is equal to ACE, a part to the whole; which is c Hyp:
abſurd. Wherefore, &c.
C
9
PRO P. XX. THE O R.
THE
HE angle at the center of the circle is double the angle at
the circumference, when the ſame arc is the baſe of both.
Let ABC be a circle, and E its center, the angle BEC, at the
center, is double the angle BAC, at the circumference; the arc
BC being the baſe of both.
For, join AE, and produce it to F; then, becauſe EA is equal
to LB, the angle EAB is equal to EBA“; but EAB, EBA, area s. r.
double EAB ; and BEF is equal to EAB, EBAb, or double EAB. b 32. 5.
For the ſame reaſon, FEC is double EAC; therefore the whole
angle BEC is double BAC. Again, let there be another angle
IDC; join EC, and produce DE to B; then the outward angle
BEC is equal to EDC, ECD, or double EDCa. For the ſame
reaſon, the angle BEF is double the angle BDF; but the whole
angle BEC is double BDC, and a part BEF is double a part
BDF; therefore the remainder FEC is double the remainder
FDC. Wherefore, &c.
PROP
ៗ
)
96
THE ELEMENTS
2
Book III.
PRO P. XXI.
THEOR.
A
Ngles that are in the ſame figment of a circle are cqual to
each other.
a
а
a
20.
b Def. 6. I.
Let ABCDE be a circle, and BAD, BED angles in the fame
ſegment BAED; theſe angles are equal.
Let F be the center of the circle ABCDE; join BF FD; theri
the angle BFD is double BAD", and likewiſe double BED;
therefore BAD, BED are equal to one another b. If the ſegment
ABDE is leſs than a ſemicircle, join AE, compleat the circle,
and, to the center F, draw AF, FE ; then the angle AFE is
double the angle ABE or ADE ~; but the angle AGB is equal
a
to EGD®; therefore the remaining angie BAD is equal to BED d.
d Cor. 32, 1. Wherefore, &c.
;
ܪ
CIS. I.
PRO P. XXII. T H E O R.
THE
TH
1
HE oppoſite angles of every quadrilateral figure inſcribed in
a circle, are equal to two right angles.
a
a 21.
و
Let ABDC be a quadrilateral figure inſcribed in the circle
ABDC, the oppoſite angles BAC, BDC are equal to two right
angles; as alſo ABD, ACD; join DA, BC.
Then in the ſegment DBAC, the angle DBC is equal to
DAC 2; for the ſame reaſon the angle BAD is equal to BCD a;
therefore the whole angle BAC is equal to the two angles DBC,
DCB; adá BDC to both; then the two angles BAC, BDC are
equal to the three angles in the triangle BDC, that is, equal to
two right angles b: But all the inward angles of any quadrila-
teral figure are equal to four right anglese; therefore the an-
gles ABD, ACD are equal to two right angles. Wherefore,
&c.
b
b 32. 1.
c Cor. 32. 1.
PRO P. XXIII. THEO R.
TWO ſimilar and uncqual ſegments of circles cannot be placed
upon the ſame right line, either on the ſame or oppoſite
fides.
For, if poſſible, let the fimilar ſegments ADB, ACB be pla-
ced upon the ſame right line AB; if not on the ſame ſide, there
can be drawn, on the fame fide, a ſegment equal to one of
them;
{
OF EUCLID.
47
them; let this be ACB; then, becauſe they are ſimilar, the an- Book IIT,
;
gle ACB is equal to ADB *; which cannot be b.. Wherefore,
a
&c.
a Det. 11.
b 16. I.
PRO P. XXIV. THE OR.
S
Imilar ſegments of circles, being upon equal right lines, are e-
qual to one another.
Let AEB, CFD be fimilar fegments, conſtitute on the equal
right lines AB, CD; they are equal to one another.
For, let the ſegment ALB be applied to the ſegment CFD,
ſo as the point A coincide with C, and B with D; then AB will
coincide with CD, and the ſegment AEB with CFD: If not,
they will cut one another, which let be in G; then the ſeg-
ment CGD cuts the ſegment CFD in the points C, G, D;
therefore a circle will cut another circle in more than two points,
which cannot be a. Wherefore. &c.
2 10.
PRO P. XXV. PRO B.
A ,
Segment of a circle being given, to deſcribe the circle
whereof it is the ſegment.
>
• Const,
If required to deſcribe the circle, whereof ABC is a ſegment,
bifect AC in D, draw DB at right angles to AC, and join AB;
3
then the angle BAD will be either equal, greater, or leſs than
the angle ABD: Firſt, let them be equal; then the fide AD is
equal to DB a, and DC to AD 5; therefore Dº is the center of a
DC 6.
the circle. 2dly, If the angle BAD is greater or leſs than ABD,C9
make the angle B.E equal to ABD, and join AE, EC ; then
d g. i.
the fide AE is equal to BE d; and becauſe AD is equal to DC,
and DE common, the angles ADE, CDE are right ones; there-
fore the ſide AE is equal to EC®; therefore the three lines EA, e 4.1.
e
EB, EC are equal; therefore E is the center of the circle if
the center E is within the ſegment, then the ſegment is greater
than a femicircle, if without, leſs; if upon the baſe of the ſeg-
ment, then it is a femicircle. Wherefore, &c.
$
.
PROP.
1
98
T H E ELE M E N T S
Book III.
PRO P. XXVI. THEO R.
Ꮲ
N equal circles, the circumferences, upon which equal angles
ſtand, ara equal to another, whether the angles are at the
center or circumferences.
In
a Def. I,
b 4. I.
Let ABC, DEF, be equal circles, and BGC, EHF equal an-
gles at the centers, and BAC, EDF at the circumferences; then
the circumference BKC is equal to ELF; join BC, EF; then,
ſince BG, GC are equal to EH, HF, and the angle BGC to
EHF, the baſe BC is equal to EF b; and becauſe the angle
BAC, is equal to EDF, and the right line BC to EF, the leg-
ment BAC is ſimilar, and equal to EDFC; but the whole cir-
cle BAC is equal to the circle EDF, and the circumference
BAC to the circumference EDF; therefore the remainder
BKC is equal to ELF. Wherefore, &c.
€ Def. 10.
and Prop.
24
PROP. XXVII. THEOR,
Ngles, that ſtand upon equal circumferences in equal circles,
are equal to each other, whether they be at the centers or
circumferences.
>
9
Let the angles at the centers of the circles ABC, DEF be
BGC, EHF, and the angles BAC, EDF, at their circumfe-
rences, ſtanding on the equal circumferences BC, EF; then the
angle BAC is equal to the angle EDF, and BGC to EHF.
For, if the angle BGC be not equal to the angle EHF, let
one of them be greater, as BGC, and make BGK equal to
EHF; then the circumference BK is equal to EF?; but EF is e-
þ Hypoth. qual to BCD; therefore, BK is equal to BC, a part to the whole,
which is impoſſible; therefore, the angle BGK is not equal to
EHF; therefore, no angle but EGC can be equal to EHF at
the center, and BAC to EDF at the circumference. Where-
fore, &c.
a 26.
1
PROP
OF EUCLID.
49
Book III.
PRO P. XXVIII. and XXIX. THEO R.
O
IN ęqual circles, equal right lines cut off equal circumferences,
the
greater equal tot he greater, and the leffer to the leffer,
and the right lines, in equal circles, which cut off equal circumfe-
rences, are equal.
I
7
с
Let ABC, DEF be equal circles, in which are the equal right
lines BC, EF, which will cut off the greater circumfcrence,
viz. BAC cqual to EDF, and the leiſer 3GC to EHF.
For, find the centers K and L of the two circles, and join
BK, KC, EL, LF; then, becauſe the two circles are cqual,
the two fides BK, KC are equal to the two ſides LL, LF ?, and a Def. s.
the baſe BC equal to EF ); therefore the angle BKC is equal to 1 Hyp.
ELF, and the circumference BGC equal to EUT; but the c 8. b.
whole circumference BGCA is equal to the whole circumſe-
rence EHFD; therefore the remaining circumſerence, BAC, is
is equal to the remaining circumference EDF. Whercfore,
&c.
And, if the circumference BGC be equal to EHF, the right
line BC will be equal to EF; for, the ſame conitruction remain-
ing, becauſe BK is equal to KC; and EL to LF2, and the an-
gle BKC to ELF 4, the bafe BC is equal to EF Whercforc, ứ 27.
ů
&c.
d
e I. 1.
PRO P. XXX. PROB.
Tout
O cut a given circumference into two equal parts.
It is required to cut the given circumference ADB into two e-
qual parts. Join AB, which biſect in Ca; from which draw a
the right line CD at right angles to ABų, and join AD, 311. I,
DB.
Now, becauſe AC is equal to CB, and CD common, the two
fides AC, CD are equal to the two fides BC, CD, and the angle
ACD to BCD b; therefore, the baſe AD is equal to DB, and ° 4.1
the circumference AD to DB4. Wherefore, &c.
b
4
ds,
G
PROP
50
THE ELEMENTS
Book III.
P R O P. XXXI.
T H E O R.
THE
HE ?ngle in a ſemicircle is a right angle, and the angle in a
legment greater than a ſemicirile is leſs than a right angle,
and the angle in a ſegment leſs than a ſemicircle, is greater than
a right angle.
a
و
a 5. I.
9
Let the angle BAC be an angle in a femicircle, ſtanding on
the diameter BC ; find the center E, and join AE; the angle
BAC will be a right angle: Let ADC be a ſegment cut off by
the right line AC, join AD, DC; then the angle ADC is great-
er than a right angle ; if the circle be compleated, the ſegment
ADC is greater than a ſemicircle, and the angle ABC in it lets
than a right angle.
For, becauſe I is the center of the circle, the angle ABE is
equal to BAE “, and the angle EAC to ACE a ; therefore,
a
the whole angle BAC is equal to the two angles ACB, ABC ;
b Cor. 32. therefore, BẠC is a right angle", and the angle BAC is great-
er than ABC; for BAE is equal to ABC. But the angles
ABC, ADC are equal to two right angles, and ABC is leſs
than a right angle; therefore ADC is greater than a right an-
gle. Wherefore, &c.
Cor. Hence the angle of a ſegment greater than a ſemicircle
is greater than a right angle, and the angle of a fegment leſs than
a femicircle, is leſs than a right angle. For the angle that the
circumference BA makes with the right line AC is greater than
a right angle; or it contins the right angle BAC. And the
angle that the circumference AC makes with the right
line AC, is leſs than a right angle; for, if BA be produced to F,
the right angle FAC contains it.
C22
)
2
a
PRO P. XXXII. THE OR.
R.
a
IF a right line touch a circle, and from the point of contact a right
line he drawn to the circle, the angles that right linc makes
with the tangent are equal to the angles in the alternate Sega
ments of the circle.
Let the right line EF touch the circle ABCD in the point B;
from any point D, in the circle, draw the right line DB; then
the angle DBF is equal to the angle in the alternate ſegment
DAB; and the angle DBE equal to DCB; for, from the point
of
OF EUCLID,
51
}
2 III.
a
0
.
1
>
of contact B, draw BA at right angles to EF'; take any roint Book III.
C in the circumference, and join AD, DC, CB.
Now, becauſe BA is drawn from B, atright angles, to EF, the 11.1.
center of the circle is in A Bb; and, becauſe ADCB is a femicircle, b 19.
the angle ADB is a right angle"; therefore ADB is equal to the c 31.
two angles DBA, DAB d; but ABF is likewiſe a right angle; d cr.32.1
therefore the angle ABF is equal to the angles DB.4, D4B;
but the angle ABF is likewife equal to the angles DBF, DKA;
therefore the angles DBA, DBF, are equal to the angles DAD,
DBA. Take the common angle DBA from both, there re- e as. 1. I.
mains the angle DBF equal to DAB, the angle in the alternate
fegment.
Likewiſe the angle DCB is equal to the angle DBE ; for,
DCB, DAB, are equal to two right angles", and DBF, DBE 6,
equal to two right angles; but DAB is proved equal to DBF;: 13. I;
therefore the remainder, DCB, is equal to DBE. Wherefore,
&c.
axI. I,
1
6 f 22.
9
PRO P. XXXIII. PROB. ·
PON a given right line to deferibe a ſegment of it. circle,
that will contain an angle equal to a giver right lined
angle.
U?
b 11. I.
C IO. I.
It is required, upon AB, to deſcribe a ſegment of a circle,
that will contain an angle equal to a given angle, C.
At the point A, with the right line AB, make the angle
BAD equal to Cº; draw AE at right angles to AD b; bifect 123. I.
LAB in F, and draw FG, at right angles, to AB, cutting AE
in the point G; join GB ; with the center G, and diſtance
GA, deſcribe the circle ABE, which will paſs through
,
the point B; for, becauſe A3 is bifected in F, and GF
drawn, at right angles, to AB, the right lines AF, FG. are
equal to BF, FG; and the angle AFG equal to BFG, therefore
;
;
AG is equal to GB d. Now, becauſe AD is a tangent to the d 4. 5.
circle', the angle BAD is equal to the angle in the alternate e 16.
fegment BEA; but the angle DAB is equal to the angle C5; 33.
Conft.
therefore the angle AEB is equal to the angle C. Wherefore,
&c.
>
e
f
f
32
و
&
PRO P. XXXIV. PROB.
T
O cut off a ſegment from a given circle that all contain an:
angle cqual to a given right lined angle.
52
Τ Η Ε Ε L Ε Μ Ε Ν Τ S
3 .
BooklII.
It is required to cut off a ſegment from the given circle ABC,
boged that ſhall contain an angle equal to the given angle D.
Draw the line EF, touching the circle in B*; from which
b. I.
draw BC, making an augle FBC cqual to the angle 1b; then
the angle TBC will be equal to the angle in the alternate feg-
ment, viz. BAC°; but FC is equal to the angle D; therefore
BAC is cqual to the angle D. Wherefore, &c.
C 32.
PRO P. XXXV. THE O R.
.
IT
F two right lines in a circle mutually cut each other, the rec-
tangle contained under the ſegment of the one, is equal to the
relagle contained under the ſogarents of the other.
a def. 15. I.
b
b. and
3.
are
C47. J.
d 5.
o
Let the two right lines AC, DB, in the circle ABCD, mu-
tually cut each other in E; then the rectangle under AE, EC,
is equal to the rectangle under DE, EB; if AC, DB, paſs
cach through the center, then the rectangle under AE, EC,
is cqual to the rectangle under DE, ED, for the lines are e-
quala
2dly, if AC, palling through the center, cut DD, 110t paſſing
through the center, at right angles, in the point E, find the
center F, and join FD ; for, becauſe BE is equal to ED 0,
the angle DEF is a right one, the ſquares of DE, EF,
equal to the ſquare of FD"; but the rectangle under AE, EC,
together with the ſquare of EF, is equal to the ſquare of FC 4,
or TD. Take the ſquare of FE, which is common, from both,
there remains the redangle under AE, EC, equal to the ſquare
of ED, that is, the rectangle under BE, ED. If the right
linc, AC, pafling through the center, cut BD, not paſling
through the center, and not at right angles, draw FG at right
couples to BD, and join FD; then BG is equal to GD";
Perectangle under BE, LD, iogeiher witi the ſquare of GE,
I cooart ihe ſquare of God. Add the ſquare ci GF to both,
equal
then the rectangle under BL, ED, with the ſquares of EG,
GF, or the ſquare of EF, are cqual to the quare of FD; but
the roangic under AE, EC, tocker with the ſquare of EF,
arc Idewilc count to the Equare of FD. Take the fquare of LF
from botn, then the rectangle zoeer AF, EC, is cqual to the
rectangle un DE, EL.
h
و
9
2
3y, If
4
O EUCLID.
53
r
3dly, If neither paſs through the center, draw GH, paſſing Book III,
through the center F, and cutting AC, BD, in E; then the
rectangle under AE, EC, is equal to the rectangle under BE,
ED; for cach is equal to the rectangle under GE, EH.
Wherefore, &c.
PRO P. XXXVI. T H E O R.
TH
IF F some point be taken without a circle, and from that point two
right lines be drawn, one of which touches the circle, and the
other cuts it, the rectangle under the whole fecunt line, and the
part between the point and convexity of the circle, is equal to the
Square of the tangeni line.
a
a
с
و
Let ABC be the circle, D the given point, and DCA, DB,
the two given right lines, of which DB touches the circle, and
DCA cuts it; the retangle under AD, DC, is equal to the
ſquare of DB.
Now, DCA either paſſes thro' the center, or not. Firſt, let it
paſs thro' the center E, and join BE; then, becauſe AC is bi.
ſected in E, and DC added, the rectangle under AD, DC, to-
gether with the ſquare of CE, are equal to the ſquare of DE ; a 6.:.
but the ſquare of DE is equal to the ſquares of DB, BED; for b 47.2.
the angle DBE is a right angle ; therefore the rectangle under 18.
AD, DC, together with the ſquare of CE, are equal to the
ſquares of DB, BE. Take the equal ſquares of BE, CE, from
both, there remains the rectangle AD, DC, equal to the ſquare
of DB.
2dly, Let DA not paſs through the center of the circle ABC;
find the center Ed, and join ED, EC, EB; draw EF, at right
angles, to AC, cutting it in F°; then AF is equal to FCf; .
therefore the rectangle under AD, DC, together with the ſquare
of CF, are equal to the ſquare of FD. Add the iquare of FS
to both; then the rectangle under AD, DC, with the iquares
of CF, FE, are equal to the ſquares of DF, FE; but the fire
of CE is equal to the ſquares of CF, Feb, and the fouard of
DE equal to the ſquares of DF, FE b; therefore the ricingle
under AD, DC, with the ſquare of CE, arc equal to the ſquare
of DL; but the facare o DE is equal to the ſquares 0: DE,
BE, therefore the rectangle de B, DC, with the ſquare
ID
of CE, are equal to the fourres ( DB, BE. Tare the caual
ſquares of BE, CE, from botla, ad the reciendole under AD: :
DC, is equal to 16 of
of D6. Therefore, 2.C.
PO??
di.
с
CIL. Ia
'
*
54
ELEMENTS
THE
Book III.
PRO P. XXXVII.
T H E O R.
j
IF, from a point without a circle two right lines be draqun, one
of which cuts the circle, and the other falls upon it ; and, if the
rectangle under the whole ſecant line, and part betwixt the point
and circle, be equal to the ſquare of the othưr line, this laſt line
ſhall be a tangent to the circle.
>
Let ſome point D, be aſſumed without the circle ABC, and
from it draw the lines DCA, DB, ſo that DCA cut the circle,
and DB fall upon it; and, if the rectangle under AD, DC, be
equal to the ſquare of DB, then DB will touch the circle in the
point B.
a 17:
bl.
C 18.
C.
d 36.
d
с Нур.
e
f 8. I.
For, let DE be drawn a tangent to the circle in the point E a;
find the center Fb, and join BF, FD, and DF.
Then the angle DEF is a right angle "; therefore the rec-
tangle under AD, DC, is equal to the ſquare of DE 4; but the
rectangle under AD, DC, is equal to the ſquare of DB €;
therefore the ſquare of DB is equal to the ſquare of DE, and
DB equal to DË; therefore the right lines DE, EF, are equal
to DB, BF; and FD common; therefore the angle DBF is
equal to the angle DEFf; but DEF is a right angle; therefore
DBF is likewiſe a right angle: Therefore DB is a tangent to
the circles. Wherefore, &c.
Cor. I. Hence, if any number of right lines, as DA, DG, be
,
drawn from the point A, cutting the circle in C and H, the
rectangles under AD, DC, and GD, DH, are equal to one a-
nother; for each of them is equal to the ſquare of BD.
II. If, from any two points in the circumference of a circle,
two tangents be drawn, ſo that, being produced, they will
meet one another; then theſe tangents will be equal to one a-
nother; for each of their ſquares, viz. of BD, DE, is equal to
the rectangle contained under AD, DC.
6 16.
a
THE
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Book II. Platev.
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Τ Η Ε
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E L M N T S
Ε L Ε Μ Ε Ν Τ S
T
OF
E U
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L I
L I
D.
во ок
IV.
D E FINITIONS.
A
1.
Right lined figure is ſaid to be inſcribed in a right lined Book IV,
figure, when every one of the angles of the inſcribed fin
gure touches every one of the ſides of the figure wherein
it is inſcribed.
II.
A right lined figure is ſaid to be deſcribed about a right lined fie
gure, when every one of the ſides of the circumſcribed figure
touches each of the angles of the right lined figure.
III.
A right lined figure is inſcribed in a circle when each of the
angles of the infcribed figure touches the circumference of
the circle.
IV.
A right lined figure is deſcribed about a circle when each of
the ſides of the circumſcribed figure touches the circumfe-
rence of the circle.
V.
A circle is inſcribed in a right lined figure, when the circum-
ference of the circle touches all the ſides of the figure in
which it is inſcribed.
VI.
A circle is deſcribed about a right lined figure when the cir-
cumference of the circle touches all the angles of the figure.
VII. A
56
TH E E L E M E N T S
BOOK IV.
VII.
n A right line is applied in a circle when its extremes are in the
circumference of the circle.
a
PRO P. I. PROB.
T'
O apply a right line in a circle, equal to a given right line, not
greater than the diameter of the circle.
It is required to apply a right line in the circle ABC, equal
to a given right line D, not greater than the diameter of the
circle.
Draw the diameter BC; if equal to D, what was required is
done; if not, the diameter BC is greater than D; put CE equal
to Da; about the center C, with the diſtance CE, deſcribe the
circle AEF; then CA is equal to CE ; but CE is equal to D;
therefore CA is equal to D. Wherefore, there is drawn,
sic,
2 3. 1.
PRO P. II.
PRO B.
IM Na given circle, to inſcribe a triangle equiangular to a given
triangle.
2 37. 3
o
.
9
It is required to infcribe a triangle, in a given circle ABC,
equiangular to a given triangle DEF: Draw the right line
GAH, touching the circle in the point A*; with the right line
AH, at the point A, make the angle HAC equal to the angle
B 23, I. DEFb, and the angle GAB equal to DFE ; join BC.
Then the angle HAC is equal to the anglc ABC; but the
angle HAC is equal to DEF; therefore, ABC is equal to DEF:
And BAG is equal to ACB «; but BAG is equal to DFE;
therefore ACB, is equal to DFE; therefore the remaining third
d Cor. 32. angles BAC, EDF d are equal; therefore the triangle ABC, is
inſcribed in the circle ABC, and equiangular to the triangle
DEF, which was required. Wherefore, &c.
, ,
32. 34
C
1
PROP
OF EUCLID.
57
Book IV.
PRO P. III. PROB.
A
BOUT a given circle, to deſcribe a triangle equiangular
to a triangle given.
و
Q
b 16.30
It is required to deſcribe a triangle about the given circle
ABC equiangular to the given triangle DEF ; produce the ſide
IF both ways to G, H; find the center of the circle K, and
draw KB any how; at the point K, with the right line KB, make,
the angles BKA, BKC, equal to the angles DEG, DFH , & 23. 1.
each to each ; at the points A, B, C, draw the right lines
LAM, MBN, LCN, tangents to the circle, in the points
A, B, C..
Then the angles that LM, MN, LN, make with the right
lines KA, KB, KC, are right angles; therefore the angles
AKB, AMB, are equal to two right angles", and equal to d 32. 1.
DEF, DEG; but BKA was made equal to DEG, therefore e 13. I.
the remainder DEF is equal to AMBf. For the ſame reaſon f Ax. 8. 31
DFE is equal to LNM, and the remaining angle MLN equal
to EDF; wherefore the triangles LMN, DEF, are equiangular.
Which was required.
C 18.
3.
d
I,
PRO P. IV. PROB.
T.
O inſcribe a circle in a given triangle.
a
а
o
a
I
b
12. I.
>
It is required to infcribe a circle in the given triangle ABC.
Biſect the angles ABC, ACB, by the right lines BD, DC a, a 9. 13
meeting each other in the point D; from which let fall DF,
DE, DG, perpendiculars, upon the right lines AB, BC, ACB.
Then, becauſe the two angles DFB, DBF, in the triangle
DBF, are equal to the two angles DEB, DBE, in the triangle
DBE, and iħe fide DB common to both ; the remaining fides
BE, ED, are equal to BF, FD, each to each For the fame c 26. j.
reaſon DF is equal to DG; therefore D is the center, and with
any of the diſtances the circle EFGd may be inſcribed in the giad 9. 3.
ven triangle ABC: Which was required.
C
PRO P. V.
PRO B.
T
O deſcribe a circle about a given triangle.
HI
1:
3.0,
1
58
T H E ELEMENTS
a
a IO, I.
Book IV. It is required to deſcribe a circle about the triangle ABC.
Bifect the ſides AB, AC, in the points D, E a; from which
draw DF, FE, at right angles to AB, AC, which will meet one
another, either within the triangle, upon one of the ſides BC, or
without the triangle, in the point F.
In either cafe, becauſe AD is equal to DB, and DF common,
the two ſides AD, DF, are equal to BD, DF; and the angle BDF
to ADFb; therefore the baſe BF is equal to the baſe AFC. For
the ſame reaſon AF is equal to FC: Therefore, if, with the
center F, and either of the diſtances AF, FB, or FC, a circle
is deſcribed, it will touch all the angles of the triangled. Which
was required,
9
b II.1.
C4. J.
d Def. 6.
PRO P. VI. PROB.
T
O infiribe a ſquare in a given circle.
a
2.
II. I.
It is required to infcribe a ſquare in the given circle ABC.
Draw the diameters BD and AC at right angles to, and bi-
ſecting each other in E ; join BA, AD, DC, CB; then ADCE
is a ſquare.
For the two fides DE, LA, are equal to DE, EA, and the
angle BEA to AED *; therefore the baſe B A is equal to AD b.
For the ſame reaſon AD is equal to DC, and DC to CB ;
therefore the four ſides are equal : But the angle BAD is a
right angle"; therefore ADC, DCB, ABC, are each right
angles"; therefore the figure is a ſquared: Which was requi-
& def. 30. 1. red.
a
b 4. I.
و
C 31. 3.
PRO P. VII. PRO B.
To deſcribe a ſquare about a given circle.
& 11. 1.
It is required to deſcribe a ſquare about the circle ABCD.
Draw the two diameters AC, BD, of the circle, cutting each
other at right anglesa; and through the points A, B, C, D,
draw FG, GH, HK, KF, tangents to the circle ABCD.
For, becauſe the angles GAE, AEB, are right angles", GF
is parallel to BD. For the ſame reaſon HK is parallel to BD;
therefore GF is parallel to HK d. For the ſame reaſon, GH is
paral-
b 16. 3
C20. I,
d 30. I.
OF EUCLID.
59
parallel to FK; therefore GHKT is a parallelogram; but GF, Book IV:
BD, are equal", and likewiſe BD equal to HK ; therefore GPU
is equal to HK f. For the ſame reaſon, GH is equal to FK; fore 34. 1.
each is equal to AC; but AC is equal to BD ; therefore GH is f Ax. 1. 1.
equal to GF; therefore the four ſides are equal ; but the angles
at G, F, are right ones; for each is equal to GAE, or FAE b; b 16. 3.
therefore all the angles are right ones; therefore GHKF is a
ſquare, deſcribed about the circle ABCD: Which was to be
done.
PRO F. VIII.
PROB.
T.
O inſcribe a circle in a given ſquare.
b
I.
ވެ
It is required to infcribe a circle in the ſquare ABCD.
Bifect the fides AB, AD, in the points E, F; through E a 10.1
draw EH parallel to AB), and through F draw FK parallel to b 31. 1.
AD or BC b; then AE, FG, are equal to one another", and C 34. 1.
likcwiſe ED, GK; but AE is equal to ED *; therefore FG is
equal to GK; buť AF is equal to AE d; therefore EG is equal d Ax. 7. Io
to FG, and FB to GH'; therefore GE, GF, GH, GK, are e.
qual; therefore if, with the center G, and either of theſe dia
ſtances, a circle is deſcribed, it will touch the ſquare in the
points E, F, H, K; for, if not, let it cut the ſquare, then a
right line drawn at right angles to the diameter of a circle will
fall within the circle; which cannot be ; therefore the circlec 16. 3.
EFHK is inſcribed in the ſquare ABCD: Which was re-
quired.
>
2
1
PRO P. IX. PRO B.
To deſcribe a circle about a given fquare.
a
It is required to deſcribe a circle about the given ſquare
ABCD.
Join AC, BD, mutually cutting each other in the point E.
For, becauſe AB is equal to AD, and AC common, ihe two
fides BA, AC, are equal to the two ſides DA, AC, and the
baſe BC equal to DC; therefore the angle BA is unio
DAC^; therefore the angle BAD is bite ted by the righe line
AÇ. For the ſame reaſon the angle ADC is biſected by : 7.
.
бо
THE ELEMENTS
.! . پ ن
Book IV. right line DB; but the angles BAD, ADC, DCB, ABC
bare equal, therefore their halfs are equal. Again, becauſe BA
is equal to AD, and AE common, and the angle BAE to
DAE, the baſe BE is equal to ED. For the ſame reaſon, AE
is equal to EC: Therefore the right lines AC, BD, are bifect.
ed in E; but the two ſides AD, DC, are equal to DC, CB b;
and the angle ADC to DCB b; therefore the baſe AC is equal
to BDC; therefore their halfs are equal : Therefore, if, with the
center E, and diſtance AE, a circle is deſcribed, it will paſs
through the points A, B, C, D,of the ſquare : Which was re-
quired. Wherefore, &c.
b
9
PRO P. X. PRO B.
T o make an iſoſceles triangle, having each of the angles at the
baſe double to the other angle.
II. 26
bi:
C 5
Cut any given right line AB in the point C, ſo that the rec-
tangle under AB, BC, be equal to the ſquare of AC"; about the
center A, and diſtance AB, deſcribe the circle BDE ; in which
apply the right line BD equal to AC, and join DA, DC;
then the triangle ADB is the iſoſceles triangle; having each of
the angles at the baſe BD double the angle at A.
For, deſcribe a circle ACD about the triangle ADC
Then, becauſe the rectangle under AB, BC, is equal to
the ſquare of AC, and BD is equal to AC, the rec-
tangle under AB, BC, is equal to the ſquare of BD; therefore
BD is a tangent to the circle ACD 4; therefore the angle
CDB is equal to the angle in the alternate fegment CAD. To
each add the angle CDA, then the whole angle ADB, or its
f s. I.
equal ABD f, is equal to the two angles CAD, CDA; but the
angle BCD is likewiſe equal to the angles CDA, CADC;
f Ax. I. 1. therefore the angle BCD is equal to CBD 8; therefore CD is
h 6.1.
equal to BD"; but BD is equal to CA; therefore CD, CA,
;
are equal 8; therefore the angle CAD is equal to CDA, but
CAD, CDA, are double CAD; therefore ABD or ADB are
each double BAD; Which was required.
d 37, 39
e 32. 3.
ܪ
h
g
A
PRO P. XI. PROB.
TO inſcribe an equilateral and equiangular pentagon in a given
OF EUCLI D.
61
с
су
It is required to inſcribe an equilateral and equiangular pen-
Book IV,
tagon in the given circle ABCDE.
Make an iſoſceles triangle FGH, having each of the angles
at the baſe GH double the other angle Fa; inſcribe the triangle a 10.
ADC in the circle ABCDE, equiangular to the triangle FGH); b 2.
then each of the angles ACD, ADC, are double the angle at A;
bifect the angles ACD, ADC, by the right lines CE, DB, 9. 3.
join AB, BC, DE, EA ; then ABCDE is the equilateral and
equiangular pentagon required.
For, becauſe the angles ACD, ADC, are bifected by the right
lines CE, DB, the five angles DAC, ADB, BDC, DCE,
ECA, are equal; therefore the five circumferences AB, BC,
CD, DE, AE, are equal d; and the right lines AB, BC, CD, d 26. 3.
DE, AE, equal to one another ®; the figure is therefore equila- € 29. 3.
teral; but it is alſo equiangular; for the circumference AB is
equal to DE. Add the circumference BCD to both, then the
whole circumference ABCD is equal to the whole circumfe.
rence EDCB ; therefore the angle AED is equal to the angle
BAEf. For the ſame reaſon, the other angles ABC, BCD, f 27. 3.
CDE, are equal to BAE, or AED; wherefore the pentagen
ABCDE is equilateral, and likewife equiangular. Which was
fequired.
d.
و
PRO P. XII. PROB.
T
O deſcribe an equilitcral and cquiangular pentagon about a
given circle.
It is required to deſcribe an equilateral and equiangular pen-
tagon about the given circle ABCDE.
Let the points A, B, C, D, E, be the angular points of an
equilateral and equiangular pentagon inſcribed in the circle;
then the circumferences AB, BC, CD, DE, EA, are equal;
draw the right lines GH, HK, KL, LM, MG, tangents to
the circle in the above points. From F, the center of thea
a
17. 3.
circle, draw to the ſame points the right lines FB, FC, FD, and
join FK, FL ; then, becauſe FB, FC, FD, are at right angles
to HK, KL, LM", the ſquares of FB, BK, are equal to the b 18. 3.
ſquare of FK , and the ſquares of FC, CK, equal to the ſquare c 47. 1.
of FK, and therefore equal to each other d; but the ſquares of d Ax. 1.80
BF, FC“, are equal; which being taken from both, the ſquare e def. 15. •
of BK is equal to the ſquare of KC; that is, BK equal to KC. For
the ſame reaſon, CL is equal to LD; and becaule BF, FK, are
equal
C
62
THE ELEMENTS
f 8. I.
g 27. 3
h . ,
Book IV. equal to CF, FK, and the baſe BK equal to KC, the angle
mBFK is equal to CFK f. For the ſame reaſon, CFL is equal to
,
LFD; but the whole angle BFC is equal to the wholc angle
h Ax: 2:1. CFD &; therefore the angle KFC is equal to the angle CFL",
8
i d. 1. and the baſe KC to CL ; but KC is equal to BK; therefore
HK is equal to KL. For the ſame reaſon, KL is equal to LM;
therefore the figure is equilatcral.
It is likewiſe equiangular ; for the angles B and C are each
right ones; and the two angles BFC, BKC, equal to two right
anglesk. For the ſame reaſon, CFD, CLD, are equal to two
right angles; but the angles CFD, CFB, are equal, therefore
CLD, BKC, are likewiſe equal. For the ſame reaſon, the
whole angle at L is equal to the angle at M; therefore the fi-
gure is equiangular; and likewiſe proved equilateral. Where.
fore, &c.
k Cor. 7
32. 1.
PRO P. XIII. P R O B.
Tº inſcribe a circle in a given equilateral and equiangular
pentagon.
a 10, I.
)
C47. I.
3
It is required to infcribe a circle in the equilateral and equis
angular pentagon ABCDE,
Biſeck the ſides BC, CD, DE, in the points H, K, L´;
from which draw the right lines HF, KF, LF, at right angles
Bil. 1.
to BC, CD, DE b; from the point F, where the right lines
HF, KF, interfect each other, draw FC, FD, FE ; then, be-
cauſe the angles FHC, FKC, are right angles, the ſquare of
FC is equal to the ſquares of FH, HC°, and likewiſe to the
ſquares of FK, KC; therefore the ſquares of FH, HC, are e-
d Ax. 1. 1. qual to the ſquares of FK, KC 4. Take the equal ſquare of HC,
CK, from both, there remains the ſquare of HF equal to the
ſquare of FK; that is, HF equal FK; therefore HF, FC, are eo
qual to FC, FK, and the baſe HC to CK; therefore the angle
HFC is equal to KFC"; but the angles FHC, FRC, are right
ones; therefore the remaining angle HCF is equal to KCF;
therefore the angle BCD is biſected by the right line FC. For
the ſame reaſon, FK is equal to FL, and the angle CDE biſecto!
by FD; therefore, becauſe FH, FK, FL, are equal, if, with
the center F, and either of theſe diſtances, a circle is deſcribed,
it will touch the ſides of the pentagon in the points G, H, K,
L, M, wherefore the circle GHKLM is inſcribed in the equila-
teral and equiangular pentagon ABCDE: Which was re*
quired.
COR
a
6 8. 1.
.
2
ܪ
1
OF É UCLID,
63
Cor. If two of the neareſt ſides of an equilateral and equian- Book IV,
gular figure be biſected, and from the point where theſe lines
cut each other, there be lines drawn to all the angles of the fi-
gure, theſe lines will biſcct all the angles of the figure.
PRO P. XIV. PRO B.
Ꮲ
Tº deſcribe a circle about a given equilateral and equiangular
pentagon.
a
a
b
9
It is required to deſcribe a circle about the equilateral and e-
quiangular pentagon ABCDE.
Bilect the right lines AB, BC , in the points Hand G;a 10. i.
a
from which draw the right lines HF, GF, interſecting each o-
ther in the point F, and at right angles, to AB, BC b; from the b 11. I.
point F draw the right lines BF, FA, FE, FD, FC, they will
bifect the angles at A, B, C, D, EC Then, becauſe AB is e-c cor. 13,
equal to AE, and Ar common, and the angle BAF equal to
E AF, the baſe BF will be equal to EFd. For the ſame reaſon, d 4. I
EF is equal to FC ; therefore, if, with the center F, and diſtance
B, E, or C, a circle is deſcribed, it will paſs through the points
A, B, C, D, E, of the equilateral and equiangular pentagon :
Which was required.
1
a
PRO P. XV.
XV. P R O B.
Tº in
inſcrive an equilateral and equiangular hexagon iiz a given
circlc.
It is required to inſcribe the equilateral and equiangular her.
agon in the given circle ABCDEF.
Draw AD a diameter to the circle ABCDEF, whoſe center is
G; with the point D as a center, and diſtance DG, deſcribe a
circle EGCH ; join EG, GC; which produce to the points
B, F; join AB, BC, CD, DE, EF, FA; then ABCDEF is
an equilateral and equiangular hexagon.
For, fince G is the center of the circle ABCDEF, GC is e-
qual a to GD; and ſince D is the center of the circle CGEH, a def15.5e
ĠD is equal to DC; therefore CGD is an equilateral triangle b; b 1. I.
but it is likewiſe equiangular. For the ſame reaſon, GDE is c cor. so fy
, se
an equilateral and equiangular triangle, and equal to the
triangle
2
#
B
THE E L E MENTS
d 15. I.
2
و
Book IV. triangle CGD; and, becauſe BG, GA, are equal to DG, GE,
and the angle BGA to DGE 4, the baſe AB is equal to DE.
For the ſame reaſon, AF is equal to CD; but CD is equal to
DE; therefore AB is equal to AF. Again, becauſe CG falls
e 13. I. upon BE, the angles CGB, CGE, are equal to two right anglesº;
but CGD, DGE, are each one third of two right angles f;
f cor. 32. i. therefore CGB is likewiſe one third of two right angles;
therefore BG, GC, are equal to CG, GF, and the angle BGC
equal to the angle CGD; therefore the baſe BC is equal to CD;
but likewiſe BG, GC, are equal to FG, GE, and the angle
BGC to FGE 4; therefore the baſe BC is equal to FE ; but BC is
proved equal to CD, and CD to DE, therefore the Gx fides
BC, CD, DE, EF, FA, AB, are equal; therefore the figure
is equilateral; it is likewiſe equiangular; for the two angles
GDC, GDE, are equal to the two angles GCD, GCB; for
each is one third of two right anglesf; therefore the whole angle
BCD, is equal to the whole CDE. For the fame reaſon, all the
other angles are equal to one another; therefore the figure is
likewiſe equiangular. Wherefore, &c.
f
2
COR. Hence the fide of a hexagon is equal to the ſemi-dia-
meter of the circle. And, if through the points A, B, C, D, E,
F, tangents to the circle, be drawn, an equilateral and equiangu-
lar hexagon will be deſcribed about the circle, as may be proved
in the ſame manner as the pentagon: And fo likewiſe a circle
may be inſcribed and deſcribed about a given hexagon.
PRO P. XVI. PRO B.
То
O inſcribe an equilateral and equiangular quindecagon in a
given circle.
IS
II.
It is required to inſcribe an equilateral and equiangular quiri-
decagon in the given circle ABCD.
Let AC be the fide of an equilateral and equiangular triangle
infcribed in the circle a, and AB the ſide of an equilateral and
equiangular pentagon, drawn from the point A ; then, if the
b;
b
circle is divided into fifteen parts, the fide of the triangle AC
will ſubtend five of them, and the ſide of the pentagon AB will
ſubtend three; therefore BC will be two of ſaid parts; therefore
biſect BC in E; BE or CE will be one fifteenth part of the cir-
1
cons
0
}
1
Book.
IV.
1
Prop
A.
Ch
Def.12.
B
0
Def.3.6) (Def.415.
ܢܝܼܝ
?
1
D
D
A
Prop 2.1
В.
I
D
Propr3
G
H
E F
Prob./4.
E
E
G
M4
IN
B4
%
1
A
13
F
F
Prop. A 5,
6
O, A
5. A
Prop.
D
E
E
B
B
F
T
B
Prok: 7
Prop. 8.
Prop.9.
Prob. 10.
G
F A
E
D
1
F
E
BH
D
DF
B
H
C
K B
H
С
Prop. 18.
G
F
A.
A
G
B
E
Prop14
Prop. 12,
BK
M
A
BH
JHL
C
KY
VI
K
Pirpa 14.
Prop. 15.
Propo
A
BA
E
D
B
H
F
F
1
1
be the laten
home
OF EUCLID.
65
{
cumference; if BC, CE, &c. be joined, the equilateral and Book IV.
equiangular quindecagon will be inſcribed : Which was requi-
red.
COR. If, from what has been ſaid of the pentagon, right
lines be drawn through the diviſions of the circle, tangents to
the ſame, an equilateral and equiangular quindecagon will be
deſcribed about the circle ; or a circle may be inſcribed or des
{cribed about the quindecagons
!
THE
is
1
7
}
1
THE
Ε Ι Ε Μ Ε Ν Τ και
È L E M N T
E
S
1
E U C
U CL L I D.
!
в оок
V.
DEFINITIONS.
А
a
1.
Book V.
PART is a magnitude of a magnitude; the leſs of the
greater, when the leſs meaſures the greater.
II.
A multiple is a magnitude of a magnitude; the greater of the
leſs, when the leſs meaſures the greater.
III.
Ratio is a certain mutual habitude of magnitudes of the fame
kind, according to quantity.
IV.
Magnitudes have proportion to each other ; which, being multi-
plied, can exceed one another.
V.
Magnitudes have the ſame ratio to each other, viz. the firſt to
the ſecond, and third to the fourth, when there are taken any
equimultiples of the firſt and third, and likewiſe any equi-
multiples of the ſecond and fourth; if the multiple of the firſt
be equal to the multiple of the ſecond, then the multiple of
the third will be equal to the multiple of the fourth; ifgreater,
greater; and, if leſs, leſs.
VI.
Magnitudes which have the ſame proportion are called Propor-
tionals,
VII:
>
OF EUCLI D.
69
a
نو
VII.
Book V.
When, of equimultiples, the multiple of the firſt exceeds the
multiple of the ſecond, but the multiple of the third does not
exceed the multiple of the fourth; the firſt to the ſecond is
;
ſaid to have a greater ratio than the third to the fourth.
VIII.
Analogy is a fimilitude of proportions.
IX.
Analogy, at leaſt, conſiſts of three terms.
X.
When three magnitudes are proportionals, the firſt has to the
third a duplicate ratio of what it has to the ſecond.
XI.
When four magnitudes are proportional, the firſt has to the
fourth a triplicate ratio of what it has to the ſecond; and
always one more in order as the proportionals ſhall be extend-
ed.
XII.
Homologous magnitudes, or magnitudes of a like ratio, are ſuch
a
whoſe antecedents are to the antecedents and conſequents to
the conſequents in the ſame ratio.
XIII.
Alternate ratio is the comparing the antecedent with the ante-
cedent, and conſequent with the conſequent.
XIV.
Inverſe ratio is, when the conſequentis taken as the antecedent,
and compared with the antecedent as a conſequent.
XV.
Compounded ratio is, when the antecedent and conſequent, ta-
ken as one, are compared with the conſequent itſelf.
XVI.
Divided ratio is, when the exceſs, by which the antecedent ex-
ceeds the conſequent, is compared with the conſequent.
XVII.
Converſe ratio is, when the antecedent is compared with the ex-
ceſs by which the antecedent exceeds the conſequent.
XVIII.
Ratio of equality is when there are taken more than two mag-
nitudes in one order, and a like number of magnitudes in ano-
ther order, comparing two to two, being in the fame ratio ;
it ſhall be in the firſt order of magnitudes, as the firſt is to the
laſt; fo, in the ſecond order of magnitudes, is the firſt to the
laſt.
XIX.
Ordinate proportion is, the ratio being, as in the laſt, as the an-
tecedent is to the conſequent, in the firſt order of magnitudes;
fo
68
Τ THE ELEMENTS
*
Book V.
m
fo is the antecedent to the conſequent in the ſecond order of
magnitudes; and as the confequent is to any other, fo is the
conſequent to any other.
XX.
Perturbate proportion is, when there are three or more magni-
tudes, and others equal to them in number, taken two and
two in the ſame ratio ; in the firſt order of magnitudes, as the
antecedent is to the conlequent; fo, in the ſecond order of
magnitudes, is the antecedent to the conſequent; and, as in
the firſt order, the conſequent is to ſome other, fo, in the le-
cond order, is ſome other to the antecedent.
m
C
A X Χ Ι Ο
M
S.
V
>
E.
I.
QUIMULTIPLES of the ſame, or of equal magni-
tudes, are equal to each other.
II.
Theſe magnitudes that have the ſame equimultiples, or whoſe
equimultiples are equal, are equal to each other,
}
PRO P. I. T H E O R.
I
F there be any number of magnitudes, equimultiples of a like
nuniber of magnitudes, each of each, whatever multiple anije
one of the former magnitudes is of its correſpondent one, the ſame
multiple are all the former magnitudes of all the latter.
1
Let AB, CD, be magnitudes, equimultiples of E, F, whate-
ver multiple AB is of E, and CD of F, the ſame multiple
AB, CD, together, is of E, F, together.
For, let the magnitudes in AB, equal to E, be AG, GB; and
the magnitudes in CD, equal to F, be CH, HD; then AG, CH,
are equal to E, F; and BG, HD, likewiſe equal to E, F; there-
;
fore, as often as AB contains E, and CD, F, fo often AB, CD,
contains E, F: Wherefore, if there are, &c.
E, :
PRO P. II. THE O R.
F the firſt be the ſame multiple of the ſecond, as the third is of
the fourth; and if the fifth be the ſame multiple of the ſecond,
that the ſixth is of the fourth; then ſhall the firſt, added to the
fifth, be the ſame multiple of the ſecond, that the third, added to the
fixth, is of the fourth.
Let
I
1
OF EUCLID.
69
Let the firſt AB be the ſame multiple of the ſecond C, that Book V.
the third DE is of the fourth F; and let the fifth BG be the
fame multiple of the ſecond c, that the ſixth EH is of the
fourth F; then AG will be the ſame multiple of C that DH is
of F.
For, becauſe AB is the fame multiple of C that DE is of F,
there are as many magnitudes in AB equal to C, as in DE, e-
qual to F. For the ſame reaſon, there are as many magnitudes
in BG equal to C, as there are in EH, equal to F; therefore
there are as many magnitudes in AG equal to C, as there are in
DH equal to F.. Wherefore, &c.
a. I.
PRO P. III. THEOR.
If the firſt be the ſame multiple of the ſecond, that the third is of
the fourth, and there be taken equimultiples of the firſt and
third, then will the magnitudes to taken be equimultiples of the
fecond and fourth.
Let the firſt A be the ſame multiple of the ſecond B that the
third C is of the fourth D; and let EF, GH, be equimultiples
of A, C then Er is the fame multiple of B, that GH is of D.
For, let the magnitudes in EF, equal to A, be FK, KE ; and the
magnitudes in GH, equal to C, be HL, LG; then there are as
many magnitudes equal to B in FE, as there are magnitudes
equal to D in GH"; wherefore FE is the fame multiple of B,
that GH is of D. Wherefore, &c.
PRO P. IV. TI E O R.
If the firſt have the ſame ratio to the ſecond that the third has to
the fourth, then shall alſo the equimultiples of the firjt have the
ſame ratio to the equimultiple of the ſecond that the equinzultiple of
the third has to that of the fourth.
C
Let there be four magnitudes, A, B, C, D, ſuch, that A is to
B as C to D. Let E, F, be taken the ſame multiples of A, C;
and G, H, the ſame multiples of B, D ; then E is to G as F
is to H. For, take K, L, any equimultiples of E, F, and
M, N, any equimultiples of G, H ; then K is the fame miul-
tiple of A that L is of Ca. For the ſame reaſon, M is the a 3:
ſame multiple of B that N is of Da; but, becauſe A is to B as
ç iş to Do if K be equal to M, I will be equal to N; if great.
.
a 3
2
i els
70
Τ Η Ε Ε L Ε Μ Ε Ν Τ S
:)
b def. s.
Book V. er, greater, and, if leſs, leſs; but K, L, are equimultiples of
E, F, and M, N, of G, H; wherefore E is to Gas F is to
HD, but it is proved, that, if K be equal to M, L is equal to
N; but, if K is equal to M, M is equal to K, and N to L; if
;
greater, greater, and, if leſs, leſs; wherefore G is to E as H is
to F. Wherefore, if four magnitudes be proportional, they
will alſo be inverſely proportional. Wherefore, &c.
PRO P. V. THE O R.
IF one magnitude be the ſame multiple of another magnitude,
that a part taken from the one is of a part taken from the other;
then the reſidue of the one ſhall be the ſame multiple of the reſidue
of the other that the whole is of the whole.
Let AB be the ſame multiple of CD, that a part taken away
AE is of a part taken away CF; then the reſidue EB ſhall be
the ſame multiple of the reſidue FD, that the whole AB is of
the whole CC.
T'or, let BE be the ſame multiple of CG that AE is of CF;
then, becauſe AE is the ſame multiple of CF that BE is of CG,
1 2
to AE will be the ſame multiple of CF that AB is of GF a; but
AE is the ſame multiple of CF that AB is of CD, and AB is
the ſame multiple of GF that it is of CD; therefore GF is e
Ax. 2. qual to CD b; take CF, which is common, from both, there re-
mains GC equal to FD; therefore AE is the ſame multiple
of CF that EB is of FD. Wherefore, &c.
b
&
PRO P. VI. THE O R.
IF
F two magnitudes be equimultiples of two magnitudes, and ſome
magnitudes, equimultiples of the fame, be taken away, the reſidue
,
ſhall either be equal to theſe magnitudes or equimultiples of them.
ܪ
For, firſt, let GB be equal to E ; if HD is not equal to F, let
KC be equal to F; then AB is the ſame multiple of E that KH.
如​: is of F a; but AB, CD, are put equimultiples of E, F; therefore
HK is the ſame multiple of F that CD is of F; therefore KH is
Ax, 2. equal to CD b. Take CH, which is common, from both, there
remains KC equal to HD; but KC was put equal to F; there- .
fore HD is equal to F; after the ſame manner it may be der
monſtrated that, if GB is any equimultiple of E, HD will be
the like equimultiple of F. Wherefore, &c.
ป
PROP
1
OF EUCLID.
71
1
PRO P. VII. THEOR.
BOOK V.
QUAL magnitudes have the ſame proportion to the ſame
magnitude, and one and the ſame magnitude has the ſame
proportion to equal magnitudes.
E
Let A, B, be two equal magnitudes, and C any third mag-
nitude, then A, B, will have the ſame proportion to C. For,
let D, E, be any equimultiples of A, B, and F any equimultiple
of C; then, if D is equal to F, E is likewiſe equal to F; if
greater, greater; and, if leſs, leſs; therefore A is to C as B is to
Ca. Again, C is to A as C is to B; for, the ſame conſtruction re- a def. s.
maining, if F is equal to D, it is likewiſe equal to E ; wherefore
C is to B as C is to A. Wherefore, &c.
ܪ .
PRO P. VIII. THEO R.
T. HE greater of two 2:nequcl magnitudes has a greater propor-
tion to ſome third magnitude than the leſs bas, and that third
magnitude has a greater proportion to the lefſer magnitude than it
has to the greater.
2
a
Let AB, C, be two unequal magnitudes, of which AB is
the greater; and let D be any third magnitude ; then AB will
have a greater proportion to D than C has to D; but D has a
greater proportion to C than it has to AB.
For, becauſe AB is greater than C, inake BE equal to C,
then AB will exceed C by AE; if AE is not greater than D, let
it be multiplied till it exceed D; and let this multiple be FG;
let GH be the ſame multiple of EB that FG is of AE; then FH
will be the ſame multiple of AB that GH is of EB?, and make a r.
K the fame multiple of C that GH is of EB; but EB is equal to
C; therefore K is equal to GHD; wherefore FH is the fame
5 ΑΣ; Το
multiple of AB that K is of C. Now, of D let L be taken a
double, M triple, and fo on, till a multiple of D is found next
greater than K, which let be N; and let M be the next multiple
of D, leſs than N; then M will not be greater than K, that is,
K will not be leſs than M; but M and D are equal to N; and
becauſe FG exceeds D, and GH is equal to K, FH will be
greater than N; that is, FH, the multiple of AB, exceeds N,
the multiple of D ; but K, the multiple of C, does not exceed N,
the multiple of D; therefore AB has to D a greater proportion . Def. z.
c . 7
than Chas to DC; but likewiſe D has to Ca greater proportion
than it has to AB; for, the fame conſtruction remaining, N,
the multiple of D, exceeds K, the multiple of C; but does not
exceed FH, the multiple of AB. Wherefore, &c.
PROT
172
THE ELEMENTS
Book. V.
PROP. IX. THEOR.
a
MAGNITUDE S which have the ſame proportion to one
and the ſame magnitude arc equal to one another; and if
magnitude has the ſame proportion to other magnitudes, theſe
magnitudes arc cqual to one another.
38,1
Let the magnitudes A, B, have the ſame proportion to Cs
then A is equal to B. If not, let A be greater or leſs than B;
if greater, then A has a greater proportion to C than B has to
Ca; but it has not; therefore A is not greater than B; if leſs,
then B has a greater proportion to C than it has to C; but it
has not“; therefore A is not leſs-than B ; and, ſince neither
greater nor leſs, it muſt be equal.
Again, if C have the ſame proportion to A that it has to
B, A is equal to B; if not, C will have a greater proportion to
the lefſer magnitude, than it has to the greater a; but it has not ;
therefore A is not greater than B, or B is not greater than A;
therefore A is equal to B. Wherefore, &c.
PRO P. X. THE OR,
O
F magnitudes having proportion to the ſame magnitude,
that which has the greater proportion to ſome third is the
greater magnitude ; and that magnitude to which the ſame has a
greater proportion is the lefjer magnitude.
37。
b 8.
Of the magnitudes A, B, if A have a greater proportion
to a third magnitude C, than B has to C, A is greater than
B; for, if not, it will be either equal or leſs. If A is equal to
B, then they would have the ſame proportion to C^; but they
have not; therefore A is not equal to B; neither is it leſs;
for then B would have a greater proportion to C than it has to
Cb; but it has not: Therefore, ſince A is neither equal nor leſs
than B, it muſt be greate
Again, if C have a greater 'proportion to B than it has to A,
then B is leſs than A; if not, let it be equal or greater; if e.
qual, then C has the ſame proportion to B that it has to A 4
but it has not; therefore B is not equal to A. If greater, then
C will have a greater proportion to A than it has to B; but it
3
has not; therefore, ſince B is not equal or greater than A, it
muſt be lefs. Wherefore, &
C
PROP
}
1
OF EUCLI D.
73
Воок у.
PRO P. XI. TIIE ON.
P
ROPORTIONS that are the ſame to any third, are the ſame
to one another.
f
L
Let A be to B, as C is to D, and C to Das E to F; then A
will be to B as E to F.
For, let G, H, K, be any equimultiples of A, C, E; and L,
M, N, any other equimultiples of B, D, F; now, becauſe 1 is
to B as C is to D; and G, H are equimultiples of A, C; and
L, M any other equimultiples of D, D; if G is equal to L, H
will be equal to M“; if greater, greater; and, if leſs, Icſs: Like- a Def. 5.
Ma
.
wiſe, becauſe C is to D as E is to l'; if H be equal to NI, K
will be equal to Na; if greater, greiter; and, if lefs, lets. Where-
fore, if G be equal to L, K will be equal to N; for they are e-
qual to H, Mb; wherefore A is to B as E to f'.
,
forc, &c.
ܪ
!
9
Where- b 4x. 1. I,
1
PRO P. XII. T H E O R.
Fany number of magnitudes be proportional, as one of the an.
tecedents is to one of the conſequents, ſo are all the antece-
dents to all the conſequents.
I
។
ny
Let the magnitudes be A, B, C, D, E, F; and, as A is to B,
ſo is C to D, and E to F; then as A is to B, fo are A, C, E,
all the antecedents, to B, D, F, all the conſequents. For, let
let G, H, K be equimultiples of A, C, E, and L, M, N be a-
other equimultiples of B, D, F; then, becauſe A is to B as
C is to D, and C to D as E to F, and G, H, K equimultiples
of A, C, E, and L, M, N any other equimultiples of B, D, F;
if G be equal to L, H will be equal to M, and K to Na; if a Def, si
greater, greater ; and, if leſs, leſs; wherefore, if G be equal to
L; G, H, K, will be equal to L, M, N; if greater, greater ;
and, if leſs, leſs b; but G; G, H, K, are equimultiples of A; A, b 1.
C, E, together; and L; L, M, N, equimultiples of B; B, D,
F, togethcr ; ſuch, that, if G be equal to L; G, H, K will be
equal to L, M, N, together; wherefore A, C, E, together, are
to B, D, F, together, as A is to Ba. Wherefore, &c.
K
PROP,
و
j
b
>
74
THE ELEMENTS
Book V.
PRO P. XIII. THEO R.
T
F the firſt is in the same proportion to the ſecond, as the third
to the fourth; and if the third has a greater proportion to
the fourth, than the fifth to the ſixth; then Jhall alſo the first have
a greater proportion to the ſecond, than the fifth to the ſixth.
3
و
ܪ
a Def. 7.
Let the firſt A have the ſame proportion to the ſecond B,
that the third C has to the fourth D; but let the third C have a
greater proportion to the fourth D, than the fifth E to the ſixth
F; then the firſt A will have a greater proportion to the ſecond
B, than the fifth E has to the ſixth F.
For, becauſe C has a greater proportion to D, then E to F;
let G, H be equimultiples of C, £; and K, L equimultiples
of D, F; ſuch, that, if G exceeds K, but H does not exceed La
and let M be the ſame multiple of A that G is of C; and N the
fame multiple of B that K is of D; then, becauſe G exceeds K,
M will exceed Nb; but G exceeds K, and H does not exceed
L; and M exceeds N, and H does not exceed L; and M, H
are equimultiples of A, L; and N, L, of B, F; wherefore A
has a greater proportion to B, than E has to F4. Wherefore, &.C.
b Dei. 5.
PRO P. XIV. TII E O R.
I F the firſt has the ſame proportion to the ſecond, that the third
has to the fourth; if the firſt be greater than the third, the
fecond will be greater than the fourth; but, if the firſt be equal to
the thiril, the ſecond will be equal to the fourth; if the firſt is leſs
than the third, the ſecond will be leſs than the fourth.
28.
Let the firſt A have the ſame proportion to the ſecond B, that
the third C bas to the fourth D; if A is greater than C, then
B is greater than D.
For, if A is greater than C, and B any third magnitude, A
has a greater proportion to B than C has io Ba; but A is to B
as C is to D, therefore C has to D a greater proportion than
C has to Bb; therefore D is leſs than B"; that is, B is greater
than D. In the ſame manner it is proved, that, if A is equal to
C, B is equal to D; and, if lets, lefs. Wherefore, &c.
2
b 13
с 10.
OF EUCLI D.
75
Book V.
PRO P. XV. THEO TN.
P4
ARTS have the ſame proportior as their like multiples, if
taken correſpondently.
Let AB be the ſame multiple of C that DE is of F; then C
will be to Fas AB is to DI.
For, let AG, GH, HB be each equal to C; and DK, KL,
LE, each equal to F; then AG, GH, HB are equal to one a•
nothera ; and likewiſe DK, KL, LE cqual to one another ; a Ax. 1, I.
therefore AG is to DK as GH is to KL, and as HB is to LEb; b 11.
therefore AB is to DE as AG is to DK; that is, as C to c 12.
F. Wherefore, &c.
ܐ .a
PRO P. XVI. T F L O R.
I ,
F four magnitudes of the ſame kind are proportional, they ſball
alſo be alternately proportional.
ܪ
Let the four magnitudes A, B, C, D, be proportional, viz. as
A is to B, fo is C to D; they will likewiſe be proportional
when taken alternately, that is, as A is to C, fo is B to D; for,
take E, F equimultiples of A, B; and G, H any equimultiples
of C and D; then, becauſe E is the fame multiple of A that T
is of B, and G the ſame multiple of C that H is of D, A is to
B as E is to Fa; but A is to B, as C is to D; therefore C is to a is.
'
Das E is to Fb; and as C is to D, fo is G to Ha; therefore
E is to F as G is to Hb; therefore, ſince E, F, G, H are four
magnitudes proportional, equimultiples of other four, A, B, C,
D, therefore, if E is equal to G, F is equal to H; if greater,
;
greater; and, if leſs, leſs; wherefore A is to C as B is to D4; c 14.
d Def. s.
Wherefore, &c.
b 11.
ܪ
.
P B O P. XVII, T H E 0 B.
IF magnitudes compounded are proportional, they ſhall alſo be
proportional
Let
76
THE ELEMENTS
2
BOOK. V.
Let the compounded magnitudes AB, BE, CD, DF, be pro-
portional; that is, let Aiš be to BE as CD is to DF; theſe
magnitudes ſhall be proportional when divided; that is, AL
ſhall be to EB as CF is tu F1).
Tor, take GH, HK, LM, MN, equimultiples of AE, EB, CF,
FD, and KX NP, any equimultiples of LB, FD ; now, be-
cauſe GH is the ſame multiple of AE that HK is of Eb; and
GH the fame multiple of AE that LM is of CF; and LM the
fame multiple of CT that MN is of FD; therefore GK is the
ſame multiple of AB that GH is of AE 4. But GH is the ſame
multiple of AE that LN is of CF; therefore GK is the fame
multiple of AB that LM is of CF.6 But LM is the ſame mul-
tiple of CF, that MN is of FD; therefore LN is the ſame mul-
tiple of CD, that LM is of CFa; therefore GK is the ſame
multiple of AB, tht LN is of CDb. But HK, MN, are the
fame multiples of UB, FD; and KN, NP any other p-
KY
quimultiples of EB, FD; wherefore HX is the fame multiple
of El that MP is of FD d. But GK, IN are equimultiples of
AB, CD; and XH, MP, any other equimultiples of EB, FD; if
GK be equal to HX, LN will be equal to MP; take HK, MN,
froin boch; then, if GH be equal to KX, LM will be equal to
NP; if greater, greater; and, if leſs, leſs; wherefore AE is to
c Def. 5.
EB as CF is to FD.. Wherefore, &c.
bil.
9
dze
و
PRO P. XVIII. T H E O R.
IF
F magnitudes divided be proportional, they ſhall alſo be propor-
tional who isipoundeil.
و
14
Let AE, EB, CF, FD, be the divided magnitudes, viz. as
AE is to EB, ſo is CF to ID, they ſhall likewiſe be proporti-
orial when compoundel, viz. as Ab is to 6 E, ſo is CD to DF;
if not, let AB be to BE as CD is to tone magnitude, either
grerter or lets than Fl); firſt, to a leſs, as DG, viz. AB to BE,
as CD to :?; therefore AE is to EB as CG is to GDa; but
AE is to EB as CF is a SD1; therefore CG is to GI) as CF
is to FDC; but CG is greater than CF; therefore DG is great-
er than I d; but it is alto leſs, which is impoflible; there- .
furt AB is not lo BE as CD to DG. In the ſame manner it is
proveri, that AB is not to BE 23 CD to one greater than DF;
berture 43 is to BE as CD is to Dr. Wherefore, &c.
b Hyp.
ܪ
с
C II,
ܪ
d 14.
PROP
OF EUCLID.
77
Book V.
PRO P. XIX. THE O R.
F the whole be to the whole, as a part taken from the one is to a a
part, taken from the other, then fall the reſidue of the one be
to the reſidue of the other, as the whole is to the whole; and if four
magnitudes be proportional, they ſhall be converſely proportional.
و
;
a 16.
а
Let the whole AB be to the whole CD, as a part taken away
AE, is to the part taken away CF; then the reſidue EB is to
the reſidue F! as the whole AB is to the whole CD; for alter-
nately as AB is to AE, fo is CD to CF2; then BE is to AE as
DF is to FCb; and BE is to DT, as AE to Cra; but as AE is b 17.
to CF, ſo is AB to CDC; therefore EB is to the reſidue FD as
с Нур.
the whole AB is to the whole CD d: Again, if AB be to BE as d 11.
CD to DF, then they ſhall be converſely proportional; for AE
is to BE as CF is to FD €; and BE is to AE as DF is to CF f
therefore as AB is to A E, fo is CD to DF8; therefore the firſt 5 18.
AB is to AE, its exceſs above the fecond, as CD, the third, is
to DF, its exceſs above the fourth h. Wherefore, &c.
,
e
e 17
; f Cor. 4.
h Def. 17.
PRO P. XX. THE OR.
I
TF there be three magnitudes, and others equal to them in num-
ber, which being taken two and two in each order, are in the
Jupne ratio ; anid if the firſi magnitude be equal to the thirit, their
the fourth will be equal to the fixth; and, if the firſt be greater
than the third, then the fourth will be greater than the ſixth; and,
if the firſt be leſs than the third, then the fourth will be leſs than
the ſixth.
3
Let A, B, C be three magnitudes, and D, E, F, others equal
to them in number; which being two in two in each order, are
in the ſame proportion, viz. A to B as D tol, and B to C as I
to F; and if the firſt A be equal to the third C, then the fourth
D ſhall be equal to the ſixth F; if greater, greater; and, if leſs,
leſs; for if A is equal to C, and B ſome other magnitude, A
has the ſame proportion to B that C hath to Ba; but A is to B a 7,
7
as D is to Eb; therefore D hath the ſame proportion to E that b Hyr,
A has to B; but B is to C as E to F; and, inverſely, C is to
B as F isto E; therefore T has to E the fame proportion that C
has
2
.
78
Τ Η Ε Ε L Ε Μ Ε Ν Τ S
ܪ
Book V. has to B; but A has the ſame proportion to B that C has to B;
therefore D has the ſame proportion to E that F has to E,
therefore D is equal to FC; if greater, greater ; and, if leſs, leſs.
Wherefore, &c.
су.
PRO P. XXI.
TH E'O R.
If there be three magnitudes, and others equal to them in num-
F
ber, which, taken two and two in each order, are in the ſame
ratio ; and, if the proportion be perturbate ; if the firſt magnitude
he greater than the third, then the fourth will be greater than
the ſixth ; and if the firſt be equal to the third, then the fourth
will be equal to the ſixth ; if leſs, leſs.
1
3
ر
>
Let the three magnitudes A, B, C, and others D, E, F, equal
to them in number, be taken two and two in the ſame ratio,
and if their analogy be perturbate, viz. as A is to B, ſo is E
to F, and B to C as D to E, and if the firſt A be greater
than the third C, then the fourth D will be greater than the
fixth F; if equal, equal; and, if leſs, leſs.
For, if A is greater than C, A has a greater ratio to B than
C has to Ba; but A is to B as E is to F; therefore E has to F
a greater ratio than C hath to B; and inverſely as C is to B,
ſo is E to D; therefore E has to F a greater ratio than E to D.
But that magnitude to which the ſame has a greater ratio, is the
lefſer magnitude; therefore F is leſs than D; that is, D is
6
greater than F; if equal, equal; and, if leſs, leſs. Wherefore,
&c,
8,
ܪ
102
و
PRO P. XXII. THE O R.
If there be any number of magnitudes, and others equal to them in
number, which, taken two and two, are in the fame ratio; then
they fall be in the ſame proportion by equality.
Let there be any number of magnitudes A, B, C, and others,
C
D, E, F, equal to them in number, which, taken two and two
in the ſame ratio, viz. A to B as D to E, and B to C as E to
F; then they ſhall be in the ſame proportion by equality; that
is, A to C, as D to F.
For, let G, H be equimultiples A, D, and K, L any equi-
multiples of B, E, and M, N any equimultiples of C, F; then,
becauſe
ز
OFEU CLI D.
70
becauſe A is to B as D to E, G is to K as His to L 4; but B is Book V.
to C as E is to F; therefore K is to M as L to N a;
Na
wherefore,
if G is equal to M, H will be equal to N; if greater, greater; a t.
4
and, if leſs, leſs; but G, M are equimultiples cf A, C and H,
N of D, F, wherefore A is to C as D to F. Wherefore, &c. e Def. s.
'
b 20.
PRO P. XXIII. T H E O R.
.
IFt there be three magnitudes, and others equal to them in number,
which, tuken two and two, are in the ſame ratio; and if their
analogy be perturbate, they fball be in the ſame proportion by equ-
lity.
Let there be three magnitudes A, B, C, and others D, E, F,
equal to them in number, which, taken two and two, are in the
fame ratio ; and if their analogy be perturbate, that is, as A is
to B, ſo is E to F, and as B is to C, ſo is D to E; then they
thall be in the ſame proportion by equality; that is, A is to Ć
as D to F.
For, let G, H, L be equimultiples of A, B, D; and K, M,
N, any equimultiples of C, E, F; then as A is to B, ſo is G to
; and as E to F, fo is M to N; but A is to B as E is to F ; a 15,
therefore G is to H as M to Nb; and, becauſe B is to Cas Db 11.
10 E, H is to K as L to M; therefore, if G is equal to K, L is e-
qual to N°; but G, K are equimultiples of A, C; and L, Neil,
of D, F; therefore A is to Cas D to T. Wherefore, &c.
ܪ
ވާ
с
c Delfi
PRO P. XXIV. THE OR,
ر
IE the first magnitude has the ſame proportion to the ſecond th.16
the third has to the fourth; and if the fifth has the ſme pro-
portion to the ſecond that the ſixth has to the fourth; then the firſt,
compounded with the fifth, ſhall have the ſame proportion to the li-
cond, that the thiril, compounded with the ſixth, l.15 to the fourth.
,
Let the firſt magnitude AB, have the ſame proportion to the
ſecond C, that the third DE has to the fourth F, and the fifth
BG have the ſame proportion to the ſecond C, that the fixth
EH has to the fourth F.
For, becauſe BG is to C as LH is to F; inverſely, C is to
BG, as F is to EH; but AB is to C as DE is to F; therefore
AB is to BG”, as DE is to EH; and AG is to GB as DH is to a :::
HEb; but as GB is to C, fo is EH to F*; therefore AG is to be
11
C as DH is to Fa. Wherefore, dic.
c Mbey
PROP.
C
86
THE ELEMENTS
Τ Ε
Book V.
PRO P. XXV. THEO R.
IF four magnitudes be proportional
, the greateſt an.1 leaſt will be
greater than the other two,
ز
Let four magnitudes AB, CD, E, F, be proportional, viz.. AB
to CD as E to F; of which let AB be the greateſt, and F the
leaft; then AB and F together, will be greater than CD and E;
for, cut off AG equal to E, and CH to F; then AB is to CD as
AG is to CH; therefore the remainder BG, will be to the re-
mainder DH, as the whole AB is to the whole DC a; but AB
is greater than CD; therefore GB is greater than HD; and,
becauſe AG is equal to E, and CH to F, then AG and F are
equal to CH and E; but BG is greater than HD; therefore
AB and F are greater than DC and E. Wherefore, &c.
a 12
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E U C L I D.
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BOOK VI.
D E F Ι Ν Ι Τ Ι Ο Ν S.
D F IN I TIO
1.
Similar right-lined figures are ſuch as have each of their fe. Book VI.
veral angles equal to one another, and the ſides about the equal
angles proportional to each other.
II.
Figures are reciprocally proportional to each other, when the
antecedent and conſequent terms of the ratio are in each figure.
III.
A right line is cut into extreme and mean ratio, when the
whole is to the greater ſegment as the greater ſegment is to the
leffer.
IV.
The altitude of any figure, is a line drawn from the vertex
perpendicular to the baſe.
V.
Ratio is ſaid to be compounded of ratios, when the ratio of
the firſt term to the laſt is produced from the quantities of the
ratios of the intermediate terms, either by multiplication, divi.
fion, or both.
1
PROP.
*
82
THE ELEMENTS
7
Book VI.
PRO P. I.
THEOR.
TRUE
1
R1.4NGLES and parallclograms that have the ſame altitude,
are to each other as their baſes.
Let the triangles ABC, ACD, and the parallelograms EB,
FD have the ſame altitude, they are to one another as their ba-
ſes ; viz. as BC to CD; for, produce BC both ways to H, M;
take BG, GH each equal to BC ; and DK, KL, LM each
,
equal to CD; and join AG, AH, AK, AL, AM;
then the triangles ABC, ABG, AGH are equal to one
a 38. r. another; and ACD, ADK, AKL, ALM equal to one
a
another a; then, becauſe HC is taken any multiple of BC,
and CM any other multiple of CD, the triangle AHC is the
ſame multiple of the triangle ABC that HC is of BC; and the
triangle CAM the ſame multiple of CAD that CM is of CD:
If HC be equal to CM, the triangle AHC will be equal to the
triangle ACM; ifgreater, greater; and if leſs, leſs; therefore ABC
Def. 5,5. is to ACD as BC is to CDb; but the parallelogram EB is double
the triangle ABC”, and FD double ACD; therefore the paral-
lelogram EB is to the parallelogram DF, as the triangle ABC is
to the triangle ACD.; therefore the parallelograms EB, DF
are to each other as their baſes BC, CD. Wherefore, &c.
ܪ
C41. I.
C
>
d
d 15. 5.
e il. 5.
PRO P. II. THEO R.
I
Fa right line be drawn parallel to one of the ſides of a triangle,
it will cut the other hides proportionally; and if a line cut the
two ſides of a triangle proportionally, that right line ſhall be pa-
rallel to the other fide of the triangle.
4
a
237. I,
by. S.
© I.
dil. S:
Let DE be drawn parallel to BC, one ſide of the triangle ABC,
then AD will be to DB as AE is to EC; for join DC, BE, then
the triangles BDE, DEC are equal"; and ADE is fome other
triangle; therefore BDE is to ADE as DEC is to ADEO: But
BDE is to ADE as BD is to ADC; and DEC is to ADE as
EC is to AEC; therefore BD is to DA as CE is to EAd; and
if BD is to DA as CE is to EA, then DE is parallel to BC.
For the ſame conſtruction remains : As BD is to DA, fo is
BDE to ADE“; and CE is to EA as DEC is to ADE; there-
fore the triangle BDE is to the triangle ADE as DEC is to
ADE d; therefore the triangles BDE, CDE are equal®; there.
foie DE is parallel to BCf. Wherefore, &c.
C
PRO P.
9. S.
f 39. 1.
OF EUCLID.
83
Book VI
PRO P. III. T H E O R.
[
F one angle of a triangle be biſeEted by a right line, which like-
wiſe cuts the oppoſite ſide, then the ſegments of that ſide will
hade the ſame proportion to one another that the other ſides of the
triangle have: And if the ſegments of that ſide have the ſame
proportion to one another that the other ſides of the triangle have,
then a right line, drawn from the point of ſection to the vertex, will
bifeat the oppoſite angle.
т
>
و
.
C
I.
;
f.
Let there be a triangle ABC, and let one of its angles, as
BAC, be bifected by the right line AD; then, as BD is to DC,
fo is BA to AC; for through C draw CE parallel to AD, a 31. s.
and produce BA till it meet CE in the point E ; then, becauſe
AC falls on the parallels DA, CE, the angle DAC is equal to
the angle ACE6: But the angle BAD is equal to CAD; b 29. 1.
therefore the angle BAD is equal to ACEd : But BAD is equal < Hyp. ,
c
to AEC 6; therefore AEC is equal to ACE d; therefore AE is d Ax. 1. 1.
equal to AC e: But BD is to DC as BA is to AEf; that is, to e 6. 1.
,
AC; and if BD is to DC as BA is to AC, then the right line
DA biſects the angle BAC.
For the ſame conſtruction remains; BD is to DC as BA is
to ACC; and BD is to DC as BA is to AEf; therefore BA is
to AC as BA is to AE 8; therefore AE is equal to ACh; there- $ 11. 5.
fore the angles ACE, AEC are equal i : But ACE is equal to 9.5.
DAC); therefore the angle AEC is equal to DAC :: But
AEC is equal to BADb; therefore BAD is equal to DACd;
therefore the angle BAC is biſected by the right line AD.
Wherefore, &c.
с
ز
h
i S. s.
ა
d
و
PRO P. IV. T H E O R.
HE fides about the equal angles of equiangular triangles
are proportional; and the ſides fubtending the equal angles
are homologous, or of like ratio.
THE
ز
Let the two equiangular triangles be ABC, DCE, viz, the
angle ACB equal to the angle DĚC; BAC to CDE; and ABC
to DCE; then the ſides that are about the equal angles are pro-
portional, and the ſides ſubtending the equal angles homolo-
gous, or of like ratio.
Let
84
THE ELEMENTS
a 17. 1.
b Cor. 17.
b
-ܛܼܲ
C 28. I.
Book VI. Let the fides BC, CE be placed in the ſame right line; then,
w becauſe the two angles ABC, ACB are leſs than two right
anglesa, and the angles ACB, DEC are equal, the angles
ABC, DEC are leſs than two right ones: The right lines AB,
DE being produced, will meet b in !ome point; which let be F;
then, becauſe the angle DCE is equal to ABC, DC is parallel to
ABC: But the angle BAC is equal to CDEL; and BAC to
d Hyp. ACD; therefore the angles CDE, ACD are equal; therefore
ED is parallel to ACf; therefore ACDF is a parallelogram,
and FD is equal to AC, and AF to DC 6; and, becauſe AC is
5
parallel to FE, "A is to AF as BC is to CEh: Alter. BA is to
BC as AF or CD is to CE. Again, becauſe " BC is to CE as
FD or AC is to DE, alter. BC is to AC as CE to ED : But AB
is to BC as CD to CE; and BC to AC as CE to ED; therefore
AB is to AC as CD to DE I, Wherefore, &c.
e
e 29. 1.
f:7. I.
$ 34 I,
si 2
h
1 12.5
PRO P. V. and VI. THE OR.
R.
If the ſides of two triangles are proportional, or if one angle of
the one be equal to one angle of the other, and the ſides about
the equal angles froportional, the triangles will be equiangular,
and 'the angles' which the homologous fides ſubtend will be e-
qual.
E
9
>
ܕ
1
2
Let there be two triangles ABC, DEF, having their fides pro-
portional; viz. AB to BC as D to EF; and as BC is to CA, fa
is ¿F to FD; and aş B A to AC, to is ED to DF; or, if the
10
angle BAC is equal to the angle ECF, and BA to AC as
TD to DF; then the triangles ABC, EDF are equiangular, and
the angle ABC equal to DEF; BCA to EFD, and BAC to
'EDF; that is, the angles which the homologous fides ſub-
tend are equal.
Tirít, Let the triangles ABC, DEF have their fides propor-
tional (fig. 1.); at the points E, F, with the right line EF; make
the angle FEG equal to the angle ABC ~; and EFG to ACB; then
the remaining angles EGF, BAC will be equal b; and becauſe
b Cor. 32. the tivo triangies ABC, EFG are equiangular, AB is to BC as
GE to EFC: But as AB is to BC to is DE to EFd; therefore
d Hyp. DE is to EF as GE is to EF €; therefore DE is equal to Guf;
for the ſame reaſon FG is equal to FDf, and EF is commin;
§ 2. Sa
therefore the triangles DEF, GEF are equal 6; and the re nain-
ing ang!es of the one equal to the remaining angles of the other,
each to cach: But the triangle EGF is equiangular to the tri-
angle ABC; therefore DEF is likewiſe equiangular to i3C.
Secondly,
223. 1.
.
@ II. 5.
6
و
8. I.
1
OF EUCLI D.
85
.
,
Secondly, Let the angle BAC be equal to the angle EDF, Book VI.
(fig. 2.) and BA to AC as ED to DF; at the point D, with the
right line DF, make the angle FDG equal to the angle EDF, or
BAC; and the angle DFG equal to the angle ACB ; then the
remaining angles at G, B, are equal h; becauſe BA is to AC as h cor, 32. 5,
ED to DF ; and likewiſe, as GD is to DF; therefore DG is
equal to DE f. For the ſame reaſon EF is equ»] to FG; and f 9. So
the triangles DEF, DFG, equiangular ; but DFG is equiangu-
lar to ABC; therefore DEF is likewiſe equiangular to ABC.
Wherefore, &c,
f
PRO P. VII. THE O R.
If there are two triangles, having one angle of the one equal
to one angle of the other, and the ſides out a ſecond angle of
the one proportional to the ſides about the correſiondent angle of
the other, and the remaining third angles, either both leſs, or both
not leſs than right angles; then the triangles will be equiangular,
and have theſe angles equal, about which the ſides are propor-
tional,
)
>
i
Let the two triangles ABC, DEF, have an angle BAC in
in the one equal to the angle EDP in the other; and the ſides
about the angles ABC, DEF, proportional, viz. AB to BC,
as DE to EF; and the other angles at C, F, either both leſs, or
both rot leſs than right angles, then the triangles ABC, DEF,
are equiangular ; the angle ABC equal to DÉF, and ACB to
DFE.
For, if the angle ABC be not equal to the angle DEF, let
one of them, as ABC, be the greater, and make the angle
ABG equal to the angle DEF“; then the remaining angles a 23. 1.
AGB, DFE), are equal, ani the ſides about the equal angles b cor 32.1,
proportional“, viz. AB to BG as DE to EF; but AB is to BC C 4.
aş DE to EF; therefore AB is to BG as AB is to BCd; there. d 11. s.
fore BG is equal to BC, and the angle BCG to BGC f; there. e 9.5.
fore BGC, BCG, are each leſs than a night angle; therefore i5.v.
DTE is leſs than a right angle 6, but BGA, BGC, are e- & ryp.
8
qual to two right angles ", and BGC is proved leſs than a right h 13. so
angle; therefore BGA is greater than a right angle ; therefore
DIE is likewite greater than a right angle, and leſs; which
is ini poſſible ; therefors ABG is not equal to DEF; nor can any
;
angle but ABC be equal to DEF. Wherefore, &c.
C
.
е
f
ܪ
g
PROP.
el
86
THE ELEMENTS
Book VI.
PRO P. VIII. THE O R.
IF
F a perpendicular be drawn in a right angled triangle, from
the right angle to the baſe, then the triangles on each Jide of the
perpcruicular will be ſimilar to the whole, and to one another.
1.
a
و
Let ABC be a right angled triangle ; and from the point A
of the right angle BC, let fall the right line AD perpendicular
to the baſe BC ; then the er angics ABD, ADC, are ſimilar to
ABC, and to one another.
For the right angles BAC, ADB, are equal; and the angle
at B common to the two triangles ABC, ABD; therefore the
a cor. 32. remaining third angles C and BAD are equal a; therefore BC is
to BA as BA is to BD b. Again, becauſe the right angles
BAC, ADC, are equal, and C common to both, the remain-
ing third angles B, DAC, are equal“; therefore BC is to AC
as AC is to DCB; they are likewiſe ſimilar to one another; for
the angles ADC, ADB, are each right angles, and the angle
C equal to BAD, and B to DAC; therefore BD is to AD as
AD is to DC; therefore the triangles ADB, ADC, are fimi-
& Def. 1.
lar to the whole", and to one another. Wherefore, &c.
COR. Hence, in a right angled triangle, if a perpendicular
is let fall from the right angle to the baſe, that perpendicular is
a mean proportional to the ſegments of the baſe; and each of the
fides containing the right angle is a mean proportional to the
whole baſe, and that ſegment next to the fide.
PRO P. IX.
PROB
T.
O cut off any part required from a given right line.
>
2 B. I.
D3i. I
Let AB be a given right line, it is required to cut off any
part of it, as one third.
From the point A draw any right line AC, making any
angle with the line AB; aſſume any point D in the line AC;
and make DE, EC, each equal AD a join BC; and through
D draw DF parallel to BC.
Then, becauſe FD is parallel to BC, a fide of the triangle
ABC, AF is to FB as AD is to DCC; but AD is one third part
of AC; therefore AF is one third part of AB; which was re-
quired. Wherefore, &c.
PROP.
OF EU C L I D.
87
Book VI,
PRO P. X.
PRO B.
То
T divide a given undivided right line, as another right line is
O
5
Let the given undivided right line be AB, and the divided
line AC, it is required to divide AB s AC is divided.
Let AC be ariy how divided in the points D, E; and making
any angle with AB; join BC, and through the points D, E,
draw DF, EGʻ, parallel to BC ; and through D draw DHK a 31. Id
parallel to AB.
Then, becauſe FH, HB, are parallelograms, their oppoſite
ſides are equal b; and, becauſe FD is parallel to GE, AT is to b 34. Je
FG as AD is to DE. Again, becauſe HE is paralle! to BC, 2
DH is to HK as DE is to 10 °; but DH is equal to FG; and
and HK to GB ; therefore DE is to EC as FG is to GB; but
AD is to DE as AF is to FG; wherefore the given undivided
line AB is cut in the ſame proportion as AC: Which was re-
quired.
PRO P. XI.
PRO B.
Two
IVO right lines being given, to find a third proportional.
Let AB, AC, be two given right lines, making any angle
with each other, it is required to find a third proportional to
them.
Produce AB, AC, to the points D, E; make BD equal to
AC a; join the points B, C; through D draw DE parallel to a 3. 1.
BC b; then AB is to BD as AC is to CE"; but BD is equal to b 31. .
AC; therefore AB is to AC as AC is to CE. Wherefore, &c. C 2.
>
7
P R O P. XII. PRO B.
HRE E right lines given, to find a fourth proportional.
,
Let A, B, C, be the three given right lines, it is required
to find a fourth proportional to them.
Let
88
THE ELEMENTS
Book VI. Let DE, EF, be two right lines, making any angle EDF
with each other; mahe DG equal to A“; GE equal to B;
and DH equal to C. Join GH; and through E draw EF paral-
lel to GHb; then DG is to GE as DH is to HF"; therefore HF
is the fourth proportional required.
a 3. I.
b 31. 1
68.
PRO P. XIII.
PRO B.
T
o find a mean proportional to two given right lines. .
Let the two given right lines AB, BC, be placed in one right
line, as AC; upon which deſcribe a ſemicircle ADC; at the point
B, draw BD at right angles to AC“; join AD, DC; then BD
is the mean proportional required.
For, becauſe the angle ADC is a right angleb, AB is to BD
as BD is to BC“; therefore BD is a mean proportional to the
given right lines AB, BC: Which was to be done.
b 31. 3.
@ 9.
PRO P. XIV. and XV. THE O R.
QUAL parallelograms and triangles, having one angle of
E
the one equal to one angle of the other, have the ſides about
the equal angles reciprocally proportional; and theſe parallelograms
and triangles that have one angle of the one equal to one angle of
the other, and the ſides about the equal angles reciprocally propor-
tional, are equal, viz. the parallelogram to the parallelogram, and
triangle to the triangle.
Let AB, BC, be equal parallelograms, and FBD, EBG, e-
qual triangles, having the angles at B equal; and let the fides
DB, BE, be in one right line, and FB, BG, in another; then
the tides DB, BE, and GB, BF, that are about the equal
angles at B, are reciprocally proportional, that is, DB is to
BE as GB to BF.
Let the equal parallelograms be AB, BC, and compleat the
parallelogrami EF; then, as the parallelogram AB is to EF, ſo
is BC to FE ~; but, as AB is to FE, fo is the baſe DB to BE b;
and, as BC is to FE, fo is GB to BF a; therefore DB is to BE
as BC is to FE
And, if DB is to BE as BG is to BF; then the parallelogram
AB is equal to BC; for, as DB is to BE, fo is AB to FED;
and, as GB is to BF, ſo is BC to FE ; therefore AB is to FE
as BC is to FE®; therefore AB is equal to BC 4.
Ses
2
a 7. 5.
bi,
с
CII, 5-
و و و و
OF EUCLI D.
89
>
C 11. 5.
Secondly, let TD, EG, be joined ; then FDB, EBG, are the e. Book VI.
.
qual triangles, and TBE
is any third magnitude ; therefore FDB
is to FBE as EBG is to FBE; but the triangle FDB is to FBE
as DB is to BE 0; and EPG is to FBE as GB is to BFD;
h 7.
therefore DB is to BE as GB is to BFC
And, if DB is to BE as GB is to BF, the triangle FDB is
equal to EBG: For, as DB is to BE, ſo is the triangle TDB
to FBE; and, as GB is to BF, ſo is the triangle EBG to FBE
therefore FDB is to FBE as EBG is to FBE; therefore the
triangle FDB is equal to EBGd; Wherćfore equal parallelo-d 9.5.
grams and triangles, &c.
PRO P. XVI. THEOR.
IF F four right lines are proportional, the rectangle contained un-
der the extremes is equal to th: rectangle under the means ;
and, if the rectangle contained under the extremes be equal to the
rcetangle contained under the means, then the four right lines are
proportional.
:
2
Lieț the four right lines AB, CD, E, F, be proportional, ſo
that AB be to CD as E is to F; then the rectangle under AB,
F, is equal to the rectangle under CD, E : For, draw AG e-
qual to F, and at right angles to AB!, and CH equal to E, 211, .
and at right angles to CD; and complet the rectangles GB,
HD: Then, becauſe AB is to CD as CH is to AG b, the rec-h 7. 5.
tangle BG is equal to HD, and if GB is equal to HD, AB is c 14.
to CD as CH is 10 AG, that is, as E to F; for the angles at C,
A, are equal, being each right ones. Wherefore, &c.
.
b
с
2
PRO P. XVII. T H E O R.
IF three right lines are proportionel, the retangle contained una
der the extremes is equal to the ſquare of the mean ; and, if
the rečtangle under the extremes be equal to the ſquare of the meani,
then the three right lines are proportionai.
Let the three right lines A, B, C, be proportional, viz. as
A is to B ſo is B to C; then the rectangle under A, C, is equal
to the ſquare of B: For, niake D equal to B, and compleat
the rectangles under A, C, and B, D; then, becauſe AC,
BD, are two rectangles, and A is to B as D is to ca, AC is a 4.5.
equal to BD); and, becauſe AC is equal to BD, A is to B as b 1.4.
M
D
a
90
THE ELEMENTS
Book VI.D is to C; but B is equal to D; therefore the rectangle under
WA, C, is equal to the ſquare of B. Wherefore, &c.
;
PRO P. XVIII.
PRO B.
UP
PCN a given right line to deſcribe a right lined figure fi-
milar and ſimilarly ſituated to a right lined figure given.
}
a
)
a
a
с •
;
Let AB be the given right line, and CDEFG the right lined
figure given; it is required upon AB to deſcribe a figure ſimilar
and timilarly ſituated to CDEFG. Join DG, DF; and, at the
points A, B, of the right line AB, make the angles BAH,
a 23. 1. ABH, equal to the angles C and CDG ?, each to each; then
1) cor. 32.8. the remaining angles AHB, CGD, will be equal b, and the
ſides about the equal angle proportional", that is, AB to BH
as CD to DG; and AH to HB as CG to GD. Again, at the
points H, B, with the right line BH, make the angles BHK,
HBK, equal to DGF, GDF, each to each ; then the remaining
third angles HKB, GFD, are equal”, and the triangles HKB,
FDG, equiangular, and the ſides about the equal angles propor-
tional. Again, make the angles BKL, KBL, equal to the
angles DFE, FDE, each to each; then the remaining third angles
at L, E, will be equal, and the ſides about the equal angles
proportional '; but all the triangles in the figure ABLKH are
proved ſimilar to all the triangles in the figure CDEFG; and,
becauſe the angles AHB, BHK, are proved equal to the two
angles CGD, DGF, each to each, the whole angle AHK is e-
qual to the angle CGF, and the ſides about the equal angles
proportional, for AH is to HB as CG to GD; and KH to HB
as FG to GD; therefore, by equality, AH is to HK as CG is
to GF. For the ſame reaſon, HK is to KL as GF to FE
and KL to LB as FE is to ED ; therefore the figure ABLKH iş
ſimilar to CDEFGd. Wherefore, &c.
b
d def. i,
P R O P. XIX. TH E O R.
SIMILAR triangles are to one another in the duplicate ratio
of their homologous ſides.
1
Let ABC, DEF, be fimilar triangles having the angles at
B and E equal; and AB, to BC, as DE to EF, and BC the
fide homologous to EF, then the triangleſ ABC to the triangle
DEF has a duplicate ratio that BC has to EF.
For,
OF EUCLI U.
91
2
>
.
7
C
For, take BG a third proportional to BC, EF”, that is, BC Book VI.
to EF as EF to BG. Join AG; then, becauſe AB is to BC
as DE to EF, alter. as AB is to DE ſo is BC to EF; but BC a 11.
is to EF as EF is to BG; therefore AB is to DE as EF is to
BGb; that is, the ſides about the equal angles B, E, of the tri- b 11. 5.
angles DEF, ABG, are reciprocally proportional; therefore e-
qual to one another; and, becauſe BC is to EF as EF is to c 14.
BG, BC has to BG a duplicate ratio of what it has to EF d; d def. 10. 5.
and, as BC is to BG fo is the triangle ABC to the triangle
ABG+; therefore the triangle ABC has to the triangle ABG a e i.
duplicate ratio of what BC has to EFb; but the triangle ABG
is equal to DET; therefore ABC is to DEF in the duplicate
ratio of BC to EF. Wherefore ſimilar triangles, &c.
Cor. Hence, if three right lines be proportional, as the firſt
is to the third, fo is a triangle deſcribed on the firſt, to a fimilar
one deſcribed on the ſecond.
° ;
PRO P. XX. THE O R.
ر
S?
IM IL A R polygons can be divided into an equal number of
ſimilar triangles, each homologous to the whole ; and polygon is
to polygon in the duplicate ratio of one homologous fide to the o-
ther.
ވެ
4
b.
+
Let ABCDE, FGHKL, be fimilar polygons, and AB, FG,
two homologous fides; join BE, EC, GL, LH; then the
number of triangles in the polygon ABCDE, are equal to the
number of triangles in the polygon FGHKL, fimilar to one a-
nother, and homologous to the whole ; and the polygon
ABCDE will be to the polygon FGHKL in the duplicate ra-
tio of the ſide AB to FG.
For, becauſe the polygon ABCDE is ſimilar to FGHKL,
the angle BAE is equal to GFL; and BA is to AE as. GF to
FL“; and the angle ABE equal to FGL b; and AB to BE as a 4.
FG to GL ; but the whole angle ABC is equal to FGH, and a
part ABE equal to FGL ; therefore the remainder EBC is equal
to LGH, and EB to BC as LG is to GH; but the angle BCD
is equal to GHK; and a part BCE to a part GHL b; therefore
the remainder ECD is equal to LHK, and the ſides about the
equal angles proportional.
Now, becauſe the triangle ABE is equiangular to the triangle
FGL, and the ſides about the equal angles proportional, the
two triangles are ſimilar, and are to one another in the duplicate
ratio
a
b 6.
;
92
THE ELEMENTS
3
C 19.
و
dil. 5
Book VI. ratio of AB to TG, or of EB to LG; but EBC is likewiſe fi-
.
milar to LGH, and are to one another in the duplicate ratio
of EB to LG, or of EC to LH; and, for the ſame reaſon,
ECD is to LHK in the duplicate ratio of CE to LH
therefore the triangles in the polygon ABCDE are equal in
number to the triangles in the polygon FGHKL, and ſimilar to
one another; therefore, becauſe the triangle ABE is to the
triangle FGL in the duplicate ratio of BE to GL; and the tri-
angle EBC to the triangle LGH, in the duplicate ratio of BE
to GL; therefore the triangles ABE, EBC, are to FGL,
LGH, as EBC is to LGHd. For the ſame reaſon, EBC, ECD,
are to LGH, LHK, as EBC is to LGH: Therefore all the an-
tecedents ABE, EBC, ECD, are to all the confequents FGL,
LHG, LHK, as ABE is to FGL '; that is, in the duplicate ra-
tio of AB to FG“; and polygon to polygon in the duplicate ra-
tio of one homologous fide to another. Wherefore, &c.
Cor. Hence, if three right lines are proportional, the poly-
gon deſcribed on the firſt is to the ſimilar polygon deſcribed on
the ſecond as the firſt is to the third; for, if X be taken a third
proportional to any two right lines, AB, FG ; then AB is to X
;
in a duplicate ratio of AB to FG; that is, any fimilar figures
deſcribed on AB, FG, are to one another in the duplicate ratio
of AB to FC.
©
C 12. $.
.
PRO P. XXI.
T H E O R.
FO
IGURES that are ſimilar to the ſame right lined figures
are alſo ſimilar to one another.
a def. 1.
Let each of the right lined figures A, B, be fimilar to the
right lined figure C; then the right lined figure A will be ſimi-
lar to the right lined figure B.
For, becauſe the right lined figure A is ſimilar to C, it is e-
quiangular to ita; and the ſides about the equal angles propor-
tional. For the ſame reaſon, B is equiangular to C, and the
ſides about the equal angles proportional; therefore each of the
figures A, B, are equiangular to C; and therefore equiangular
to one anotherb, and the ſides about the equal angles proportion-
al'; wherefore A is fimilar to B. Wherefore, &c.
9
bax. I. 1.
9
CII. S
។
PROP .
OF EUCLID.
93
1
1
PRO P. XXII. THE O R.
Book VI.
IF F four right lines are proportional, the right lined figures fimi-
lar, and ſimilarly deſcribed upon them, are proportional; and, if
fimilar right lined figures ſimilarly deſcribed upon right lines be
proportional, the right lines shall alſo be proportional.
a
a
)
с
Let four right lines AB, CD, EF, GH, be proportional,
viz. as AB is to CD fo is EF to GH; on AB, CD, let the fi-
milar figures KAB, LCD, be ſimilarly deſcribed“; and upon a 18.
EF, GH, let MF, NH, be deſcribed ſimilar to one another;
then KAB will be to LCD as MF is to NH.
For, to AB, CD, take X a third proportional”, and O, a third b Ir.
proportional to EF, GH. Now, becauſe AB is to CD as EF is
,
to GH, and CD is to X as GH is to O, then AB is to X as EF
is to O'; but, as AB is to X, fo is the right lined figure KAB C 22. s.
to the ſimilar figure LCD d; and, as EF is to O, fo is MF to'd cor. 200
NH 4; therefore, as the right lined figure KAB is to the ſimilar
figure LCD, ſo is the right lined figure MF to the ſimilar
figure NH"; and, if KAB is to LCD as MF is to NH, then e 11. s.
AB is to CD as EF is to GH.
For, if not, let AB be to CD as EF is to PR £; upon PR de- f 12.
ſcribe a figure SR ſimilar to MF or NH; then KAB is to LCD
as MF is to SR, and as MF is to NH ; therefore SR, NH, have
the ſame proportion to MF; therefore SR is equal to NHS, and $ 9. 5.
alſo ſimilar to it; therefore PR is equal to GH; therefore AB
is to CD as EF is to GH. Wherefore, &c.
f
ز
PRO P. XXIII. THEO R.
E
QUIANGULAR parallelograms have the proportion to
one another that is compounded of their ſides.
a
2
Let AC, CF, be equiangular parallelograms, having the
angle BCD equal to the angle ECG; then the parallelogram
AC, to the parallelogram CF, is in the proportion compounded
of their ſides, viz. of BC to CG, and DC to CE; for, place
BC in a right line with CG, and DC in a right line with
CE“, and compleat the parallelogram DG; then, as BC is a 14. L.
to CG, ſo let K be to L; and, as DC is to CE, fo let L be
to Mb; but the ratio of K to M is compounded of the ra-b 12.
tios of K to L, and L to MC; therefore the ratio of K to M is c def. So
that compounded of BC to CG, and DC to CE ; but BC is to
CG as AC is to DG d; and DC is to CE as DG is to CF4;d so
but BC is to CG as K to L, and DC to CE as L to M; therefore
AC
1
94
THE ELEMENTS
Book VI. AC is to CF as K to M°; that is, as BC to CG, and DC to
.
CE. Wherefore, &c,
.
C 22. 5.
PRO P. XXIV. THEO R.
IN every parallelogram the parallelograms that are about the di-
ameter are ſimilar to the whole, and alſo to one another.
a
29. I.
b 46
a
In the parallelogram ABCD the parallelograms GE, KH,
are ſimilar to the whole ABCD, and likewiſe to one another.
For, in the triangles ADC, AGF, the angle AGF is equal to
the angle ADC, and the angle AFG to ACD “; and the angle:
GAF common to both; therefore the triangles AGF, ADC,
are equiangular. For the ſame reaſon, the triangles AEF,
ACB, are equiangular, and the ſides about the equiangles pro-
portional b. Again, in the triangles AGF, FKC, the angle
GAF is equal to the angle KFG*, and AGF equal to FKČ;
for each are equal to the angle ADC , and AFG to FCK;
a
therefore the triangles AGF, FKC, are equiangular, and the
fides about the equal angles proportional b. For the ſame rea-
ſon, AEF is equiangular to FCH; and the ſides about the equal
angles proportional. Then, becauſe the two angles KFC, HFC,
are equal to the two angles GAF, EAF; that is, the whole
angle KFH equal to the whole angle GAE, and the angle C
common to both; and the angles at K, H, equal to the angles
at D, B, each to each, the parallelogram KH is equiangular to
DB; for the ſame reaſon GE is equiangular to DB; therefore
KH is equiangular to GE; and the ſides about the equal angles
proportional : For, becauſe the angles GAE, KFH, are equal,
GA is to AE as KF is to FH; but KF is to FH as DA is to
AB, for each are proportional to AF, FC°. For the ſame rea-
ſon, the ſides about the other angles are likewiſe proportional;
therefore the parallelogram DB is fimilar to KH; but GE is like-
wiſe ſimilar to KH; therefore GE is ſimilar to DB. Where-
fore, &c.
C. II. S.
PRO P. XXV. PRO B.
Tdeſcribe a figure ſimilar to a given right lined
figure, and e.
qual to another given right lined figure.
Let ABC and D, be two given right lined figures; it is required
to deſcribe a right lined figure fimilar to ABC, and equal to D.
On
OF EUCLI D.
95
a
b
& 29. I.
d 13
€
On the ſide BC, of the given figure ABC, make a parallelo- Book VI,
gram, BE, equal to it“; and on the ſide CE make the paralle-
, a
logram CM equal to the right lined figure D b; and the angle a 42.1.
FČE equal to the angle CBLb; then BC, CF, as alſo LE, EM, 44. I.
will be right lines. Find GH a mean ſproportional to BC,
CFd; and on GH deſcribe the right lined figure KGH ſimilar © 14. 1.
and alike ſituate to ABC'; then, becauſe BC is to GH as GH
is to CF; and, as BC is to CF, ſo is the right lined figure ABC 18.
to the right lined figure KGHf; but, as BC is to CF, ſo is the
f
parallelograin BE to EF 5; therefore, as the right lined figure f Cor. 20,
ABC is to the right lined figure KGH, fo is the parallelogram & I.
BE to the parallelogram EF h, altern. as ABC is to BE, ſo is
KGH to EF: But the right lined figure ABC is equal to the
parallelogram BE; therefore the right lined figure KGH is equal
to the parallelogram FE'; but FE is equal to the right lined fi-
gure D; therefore the right lined figure KGH is equal to D:
Which was required.
g
h II. S.
i 14. S.
PRO P. XXVI. THE O R.
IF, in a parallelogram, be conftitute another parallelogram Fimi-
lar to the whole, and alike ſituate, and having an angle com-
mion with it, they ſhall be about the ſame diameter.
>
Let the parallelogram AF be conſtitute in the parallelogram
ABCD, ſimilar to it, and alike fituate, having the angle ĎAB
common to both ; then the parallelograms ABCD and AF are a-
bout the ſame diameter AC.
For, if not, let AHC be the diameter of the parallelogram
BD; and produce GF to H; draw HK parallel to AD or BC;
then, becauſe the parallelograms ABCD, KG, are about the
ſame diameter, they will be ſimilar to one another“; and DA to
AB as GA to KAD; but, becauſe the parallelograms ABCD,
Gf, are likewiſe fimilar, DA is to AB as GA is to AE
с Нур.
therefore, as GA is to AE ſo is GA' to AK 4; therefore AE is
equal to AK®, the greater to the leſs, which is impoflible
therefore the parallelograms AH, ABCD, are not about the
fame diameter AHC; therefore no other but AF can be about the
fame diameter with ABCD. Wherefore, &c.
a
b
a 24.
3
d.
d II. 5
ز
e 9. 5
PROP
96
THE ELEMENTS
Book VI.
PRO P. XXVII. THE O R.
O F all parallelograms applied to the ſame right line, and wani-
ing in figure by parallelograms ſimilar and alike ſituate to that
deſcribed on half the line, the greateſt is that which is applied to
the half line, and ſimilar to the defect.
Let AB be a right line bifected in the point C; and let the
parallelogram AD be applied to the right line AB, wanting in
figure by the parallelogram CE, ſimilar and alike fituate to that
deſcribed on half the line AB; then AD is greater than a paralle-
logram applied to any other part of the right line AB, wanting in
figure by a parallelogram ſimilar and alike fituate to CE, For, let
the parallelogram AF be applied to the right line AB, wanting in
figure by the parallelogram HK, ſimilar and alike ſituate to ČE;
then the parallelogram AD is greater than AF. For, becauſe the
parallelogram CE is ſimilar to HK, they will ſtand about the ſame
,
diameter. Let DB, that diameter,' be drawn, and the figure
a
deſcribed ; then the parallelograms CF, FE, are equal b; add
HK, which is common to both; then the whole CH is equal to
the whole KE ; but CH is equal to GC°; add CF, which is
common; then the whole AF is equal to the gnomon EKN ;
but the parallelogram CE is greater than the gnomon EKN ;
therefore CE, that is, AD, is greater than AF. Wherefore, Sic.
a 26.
b 43, 1.
દ રૂ6, 1,
PRO P. XXVIII. PROB.
O
U?
T
a
PON a given right line to apply a parallelogram equal to a
given right lined figure, and deficient by a parallelagram
fimilar to a given parallelogram; but the right lined figure to,
which the parallelogram is to be made equal, muſt not be greater
than that deſcribed on half the line, as the defeet muſt be ſimilar.
It is required, upon the given right line AB, to apply a paral-
lelogram equal to the right lined figure C, and deficient by a
parallelogram ſimilar to D; and the right lined figure C not
greater than the parallelogram deſcribed on half the line AB,
which is ſimilar to D. For, biſect AB in E, and on EB de-
ſcribe a parallelogram EF fimilar and alike ſituate to Da, and
compleat the parallelogram AG.
Now, AG is either equal or greater than C; if equal, what
was required is done. If not, make the parallelogram KLMN
ſimilar
R 18.
OF EUCLID.
97
a
ز
timilar and alike ſituate to D, and equal to the exceſs by which Book VI.
EF exceeds Cb; then EF is equal to C, and KLMN together; u
therefore KLMN is leſs than EF; and, becauſe they are ſimilar, the a 18.
ſide GF is greater than LM, and GE than LK; make GO equal b 25.
to LM, and GX to LK; and compleat the parallelogram GP,
which will be ſimilar to, and about the ſame diameter with
EF“; let this diameter GB be drawn, and produce XP to R, and © 26.
OP to S; then TS will be equal to C, and wanting in figure by
SR, which is ſimilar to D4: Which was to be done.
d 236
PRO P. XXIX.
PRO B.
То O apply a parallelogram upon a given right line, equal to a gi-
ven right lined figure, exceeding by a parallelogram ſimilar to
another given parallelogram.
a
Б
b 25
2
Upon the given right line AB, it is required to apply a paral-
lelogram equal to the given right lined figure C, exceeding by
a parallelogram ſimilar to the given parallelogram D.
Bifect AB in E; upon EB deſcribe the parallelogram EL, fi-
.
milar and alike ſituate to D *; and the parallelogram GH equal a 18.
to C and EL together b, and ſimilar and alike ſituate to Da;
let KH be a fide homologous to FL, and KG to FE; then, be-
cauſe the parallelogram GH is greater than the parallelogram
EL, the right line KH is greater than PL, and í G than FE:
Produce FL and FE to M and N, ſo that FM be equal to KH,
and FN to KG; compleat the parallelogram NM; then MN is
equal and ſimilar to GH ; but GH is fimilar to ÉL; therefore
MN is ſimilar to EL"; therefore EL is about the ſame diameter C 21.
with MN d; let FX, their diameter, be drawn, and deſcribe the d 26.
figure : Then, fince GH is equal to EL and C together, as alſo
to MN; therefore MN is equal to EL and C together.
Take KL, which is common, from both; then the gnomon
MPE is equal to C; and, becauſe the parallelograms AN, NB,
е
are equal , AN is equal to LOf; and, if BX be added, AX is f 3. and
equal to the gnomon MPE; therefore AX is equal to C. Where- Ax. 1. 1.
fore, &c.
C
و
ܪ
>
PRO P. XXX
PRO B,
Το ει
O cut a given right line into extreme and mean ratio.
}
N
N
1
98
THE ELEMENTS
1
a 46. I.
-
b 22.
Book VI. It is required to cut the given right line AB into extreme and
.
pamband mean ratio.
Upon AB deſcribe the ſquare BC, and to AC apply the pa-
rallelogram CD, equal to the ſquare BC, exceeding by the fi-
gure ADb, ſimilar to BC; but BC is a ſquare ; therefore AD is
alſo a ſquare. From the equal parallelograms BC, CD, take a-
way the common parallelogram CE; then the remainder BF will
be equal to AD; but BF is equal to AD; therefore FE is to ED
| ;
as AE is to EB"; that is, AB is to AE as AE is to EB: Or, let
AB be cut in E, ſo that the rectangle under AB, BE, be equal
to the ſquare of AEd Wherefore, &c.
C 14
II.2.
.
PRO P. XXXI. THEO R.
IN every right angled triangle, any figure deſcribed upon the ſide
ſubtending the right angle, is equal to the two ſimilar figures de-
fcribed upon the ſides containing the right angle.
.
a
a
Let ABC be the right angled triangle, the figure deſcribed on
BC, ſubtending the right angle, is equal to the two fimilar
figures deſcribed on BA, AC; for, from the point A, let fall
the perpendicular AD, then the triangle ABC is divided into
;
the two fimilar triangles ADB, ADC, then, becauſe the tri-
angle ABC is ſimilar to the triangle ABD, CB is to BA as BA
is to BD a, and CB is to BD as the figure deſcribed on CB is to
b Cor. 20, the ſimilar figure deſcribed on BA 6.
For the ſame reaſon, as
BC is to CD, fo is the figure deſcribed on BC to the ſimilar one
deſcribed on AC: Wherefore, as BC is to BD, and DC toge-
ther, fo is the figure deſcribed on BC to the two ſimilar figures
6 24. S. deſcribed on BA, ACS, together ; but BC is equal to BD, and
DC together; therefore the figure deſcribed on BC is equal
to the two ſimilar figures deſcribed on BA, AC. Wherefore,
&c.
PRO P. XXXII. THE O R.
IF two triangles having two fides proportional to two ſides, be
ſo compounded or ſet together at one angle, that their homolo-
gous fides be parallel; then the other ſides of theſe triangles will
be in one right line.
If the triangles ABC, DCE be fo placed at the point C, that
, the fide DE be parallel to AC, and ỘC to AB; then BCE will
be a right line.
For;
OF EUCLI D.
99
a
០
For, becauſe the homologous fides AB, DC, are parallel, and Book VI.
AC falls upon them, the alternate angles BAC, ACD, are e.
qual'; for the ſame reaſon, CDE is equal to ACD; then, a 29. 1.
ſince the two triangles BAC, CDE, have the angles at A
and D equal, and the ſides about them proportional, viz. BA
to AC as CD to DE, the triangles are equiangular, viz. theb 6.
angle ABC equal to DCE, and ACB to DEC; but the angle
ACD is proved equal to BAC; therefore, the whole angle
ACE is equal to the two angles ABC, BAC. Add the common
angle ACB to both, then the two angles ACE, ACB, are equal
to the three angles ABC, ACB, BAC; that is, equal to two
right angles"; therefore BCE is one right lined. Wherefore,c 32. I.
&c.
d 14. I.
PRO P. XXXIII. T H E O R.
IN N equal circles, the angles are in the ſamne proportion to one an-
other as the circumferences on which they ftand, whether the
angles be at the centres or the circumference; ſo likewiſe are
foetors, as being at the centres.
ر
2
Let ABC, DEF, be equal circles, and the angles BGC,
EHF, at the centres G, H, and BAC, EDF, angles at their
circumferences; then the angle BGC will be to the angle EHF
as the circumference BC is to the circumference EF; and like-
wiſe the angle BAC to the angle EDF, and the ſector BGC to
the ſector EHF, as the circumference BC to EF. For, take
any number of circumferences, as CK, KL, each equal to BC;
and any number of circumferences, as FM, MN, each equal
to EF; join GK, GL, HM, HN; then, becauſe the circum-
ferences BC, CE, KL, are equal, the angles BGC, CGK,
KGL, are likewiſe equal“; therefore, BL is the ſame multiple a 24. 3.
a
of BC, that the angle BGL is of the angle BGC; for the ſame
reaſon, EN is the ſame multiple of EF, that EHN is of EHF ';
therefore, if the circumference BL be equal to the circum-
ference EN, the angle BGL is equal to the angle EHN, if
greater greater, and if leſs leſs; therefore, as BC is to EF, fo
is BGC to EHFb; and ſo is BAC to EDFC. Again, as the b def. 5. 5.
ss
circumference BC is to EF, fo is the ſector BGC to the ſector? 15. s.and
EHF; for, join BC, CK, EF, FM, and aſſume the points X,
0, in the circumference BC, CK, and join BX, XC, CO,
OK ; then, becauſe BG, GC, are equal to CG, GK, and
contain equal angles, the baſe BC is equal to the baſe CK 4,0 4. I.
d
and the triangles equal ; and, becauſe the right line BC is és
qual
20. 3.
ܪ
E-
*
1
100
THE ELEMENTS
с
ez 8 3
Book VI. qual to the right line CK, the circumference BXC is equal to
the circumference COK C: therefore the angle BXC is equal to
the angle COK: Therefore, the ſegments BXC, COK, are equal
f 24. 3. and and ſimilar f; but the triangles BGC, CGK, are equal; there-
det. 11. 3. fore the whole ſector BGCX is equal to the whole ſector
CGKO; in the ſame manner the ſectors EHF, FHM, are pro-
ved equal; therefore BK is the ſame multiple of BC, that BGK
is of BGCX; and EM the fame multiple of EF that the ſector
EHMF is of the ſector EHF; therefore, if the circumference
BK is equal to the circumference EM, the ſector BGK is equal
to the ſector CHM, if greater, greater, and if leſs, leſs; there-
b def. 5. 5. fore, as BC is to EF, ſo is the ſector BGCX to the ſector EHFb.
Wherefore, &c.
Cor. I. An angle at the centre of a circle is to four right
angles, as the arch on which it ſtands is to the whole circum-
ference; for, as the angle BAC is to a right angle, fo is the
arch BC to a quadrant, the confequents quadrupled; then BAC
is to four right angles as BC is to the whole circumference.
COR. II. The arches IL, BC, of unequal circles, which ſub-
tend equal angles, whether at the centres or circumferences, are
ſimilar : For IL is to the whole circumference ILE as the angle
IAL, or BAC, is to four right angles; and ſo is the arch BC to
the whole circumference BCF; therefore the arches IL, BC,
are ſimilar.
COR. III. Two femidiameters AB, AC, cut off ſimilar arch-
es IL, BC, from concentric circumferences.
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THE
Τ Η Ε
Ε L Ε Μ Ε Ν Τ S
1
F
EU CL I D.
LID.
BOOK
XI.
D E F 1 Ν Ι Τ Ι Ο Ν S.
Book XI;
A , ,
a
I.
Solid, is that which hath length, breadth, and thick-
neſs.
II.
The term of a ſolid, is a ſuperficies.
III.
A right line is perpendicular to a plain, when it makes right
angles with all the lines that touch it, and are drawn in the
ſame plain.
IV.
A plain is perpendicular to a plain, when all the right lines in
one plain, drawn at right angles to the common ſection of the
two plains, are at right angles to the other plain.
V.
The inclination of a right line to a plain, is the acute angle
contained under that line, and another right one drawn in
the plain, from that end of the inclining line, which is in
the plain, to the point where a right line falls from the other
end of the inclining line, perpendicular to the plain.
VI,
το2
Τ Η Ε Ε Ι Ε Μ Ε Ν Τ S
Book XI.
VI.
The inclination of a plain to a plain, is the acute angle con-
tained by the right lines drawn in both plains, to the fame
point of their common ſection, and making right angles with
it.
VII.
Plains are inclined ſimilarly, when their angles of inclination
are equal.
VIII.
Parallel plains are ſuch, which being produced, never meet.
IX.
Similar ſolid figures are ſuch as are contained under an e-
qual number of ſimilar plains.
X.
Equal and fimilar ſolid figures are ſuch as are contained by an
equal number of ſimilar and equal plains.
XI.
A ſolid angle is the inclination of more than two right lines
that meet in one point, but are not in the ſame ſuperficies.
XII.
A pyramid is a ſolid figure, contained by more than two
plains ſet upon one plain, and meeting at one point in the
vertex.
XIII,
A prifm is a folid figure contained by plains, whereof the two
oppoſite are equal, ſimilar, and parallel ; and the other pa-
rallelogranis.
XIV.
A fphere is a ſolid figure, deſcribed by a ſemicircle revolving
about its diameter, which remains fixed in the ſame pofi-
tion.
XV.
The axis of a ſphere is that fixed right line about which the
femicircle revolves,
XVI.
The centre of a ſphere is the ſame with that of the ſemi-
circle.
XVH.
The diameter of a ſphere is a right line drawn through the
centre, and terminated on either lide by the ſuperficies of the
ſphere.
XVIII.
A cone is a folid figure deſcribed by a right angled triangle
revolving about one of the fides, containing the right angle,
remaining fixed. If the fixed right line be equal to the o-
ther fide containing the right angle, then it is a rectangular
cone;
1
OF EUCLI U.
IO3
;
a
cone; if leſs, an obtufe angled cone; and if greater, an a- Book XI.
cute angled cone:
XIX,
The axis of a cone is that fixed right line about which the tri-
angle is moved.
XX.
The baſe of a cone is the circle deſcribed by the revolving line.
XXI.
A cylinder is, a figure deſcribed by a right angled parallelogram,
revolving about one of the ſides, containing the right angle,
remaining fixed.
XXII.
The axis of a cylinder is that fixed right line about which the
parallelogram is moved.
XXIII.
The baſes of a cylinder are the circles deſcribed by the motion
of the two oppoſite ſides of the parallelogram.
XXIV.
Similar cones and cylinders are ſuch, whoſe axes and diameters
of their bafes are proportional.
XXV
A cube is a folid figure contained by fix equal ſquares.
XXVI.
A tetrahedron is a ſolid figure contained by four equal equilateral
triangles.
XXVII.
An octahedron is a ſolid figure contained by eight equilateral
triangles.
XXVIII.
A dodecahedron is a folid figure contained by twelve equal equi-
lateral and equiangular pentagons.
XXIX.
An icoſahedron is a ſolid figure contained by twenty equal e-
quilateral triangles.
XXX.
A parallelopipedon is a ſolid figure contained by fix quadrilate-
ral figures, whereof thoſe that are oppoſite are parallel.
a
a
PRO P. I.
THE OR.
O
NE part of a right line cannot be in a plain fuperficies, and
another part above it.
For,
104
THE ELEMENTS
a
Book XI. For, if poflible, let the part AB of the right line ABC be in a
plain ſuperficies, and the part BC above the ſame; there will be
ſome right line in that plain which will make one right line
with AB, which let be DB ; then the two right lines ABC,
ABD, will have one common ſegment AB; which is impoſ-
a Cor. 14.8. fiblea. Wherefore, &c.
PRO P. II. T H E O R.
IF two right lines cut each other, they are both in one plain, and
every triangle is in one plain.
Let the two right lines AB, CD, cut each other in the point
I, they are both in one plain.
For, take any points F, G, in the right lines AB, CD, and
join CB; then the right lines AB, CD, are in one plain, and
the triangle ECB is in one plain. For the parts DF, AG,
cannot be in one plain, and FC, GB, above it a; therefore DC,
AB, are in one plain ; and, becauſe the points B, C, are in one
def, 4.11. plain b, therefore the triangle ECB is in one plain. Where-
,
fore, &c.
1
I.
)
PRO P. III. THE O R.
IF two plains cut each other, their common feflion will be a right
line
3
Let the two plains be AB, BC, cutting each other; and let
BD be their common ſection; then BD is a right line. For,
if not, let the right line BED be drawn in the plain CB, and
BFD in the plain BA; then two right lines bound a figure ;
Ax. 10, I.
which cannot be a. Wherefore, &c.
a
>
PRO P. IV. THEO R.
IF F a right line ſtand in the common ſection of two right lines,
cutting one another, and at right angles to the ſame ; then it
Shall be at right angles to the plain paſſing through theſe lines.
;
Let
ma
OF EV CL I D.
105
a
ܪ
و
a
Let the right line EF ſtand in the common ſection at right
Book XI,
angles to the two right lines AB, CD; then EF is likewiſe at
right angles to the plain paling through AB, CD. For, take
the right lines AE, ED, EB, EC, equal to one another, and
join AD, BC; through E draw GEH to the points G, H, in
the right lines AD, BC ; join FD, FC, FA, FB, FG, FH;
then, becauſe the two ſides AE, ED, are equal to the two ſides
BE, EC ; and the angles AED, BEC, equal; the baſe AD a 15. 1.
is equal to the baſe ŽC ; and the remaining angles EAD,
EDA, equal to the angles EBC, ECB, each to each b; but the b 4. I,
two angles AEG, EAG, in the triangle AGE, are equal to
the two angles BEH, EBH, in the triangle HBE ; and a ſide
AE in the one equal to a fide EB in the other; the remaining
fides AG, GE, in the one are equal to BH, HE, in the other,
each to each"; but, becauſe AE is equal to 'EB, and TE com-C 25. 1.
mon, and at right angles to AB, the baſe AF is equal to the
baſe FB. For the fame reaſon, FD is equal to TC; but FA,
AD, are proved equal to FB, BC, each to each; and the baſe
FD equal to FC; therefore the angle FAD is equal to the angle
FBC4 Again, becauſe FA, AG, are proved equal to FB, d 8. I,
.
BH, each to each, and the angle FAG equal to FBH ; the baſe
FG is equal to FH 6. Now, ſince FE, EH, are equal to FE,
EG, and the baſe FH equal to FG; the angle FEH is equal to
FEG; therefore each is a right angle"; therefore FE is at right e def. 10. 8.
angles to all the lines paſſing through AB, BC; and therefore at
right angles to the plain palling through AB, DCf. Where- f def. 3.
fore, &ic.
ز
PRO P. V. THE O R.
IF
Fa right line ſtand in the common ſection of three right lines, and
at right angles to them, theſe thrce right lines ſhall be in the
ſame plain.
1
Let the right line AB ſtand at right angles to the three right
lines BC, BD, BE, in the point of contact B; theſe three lines
ſhall be in the ſame plain.
For, if not, let BD, BE, be in the fame plain, and BC above
it; and let the plain paflug through AB, BC, be produced, till
it meet the plain paſſing through BD, BE; and let BF be their
common ſection, then BF is a right line"; then the three right a 3.
lines BE, BD, BF, are in one plain; but AB is at right angles
to DD, BE; therefore at right angles to BF, meeting BD, BE,
in B ; but the angle ABC is a right angle“; and the angles b 4.
O
ABC, c byp:
a
b
a
с
106
THE ELEMENTS
Book XI. ABC, ABF, are in the fame plain d; therefore the angle ABC
is equal to ABF, a part to the whole; which is impoſſible ;
d def. 3. therefore BC is in the fame plain with BD, BE. Wherefore,
&c.
PRO P. VI. THE O R.
á
I there be two parallel lines, and
a point taken in each of them;
the right line joining theſe points fball be in the ſame plain with
the parallel lines.
W
Let AB, CD, be two parallel lines, and E, F, points taken
in them ; then the right line EF joining theſe points is in the
ſame plain with the parallels; if not, let it be elevated above the
plain, as EGF; through which let ſome plain be drawn, whoſe
common ſection with the plain in which the parallels are, let be
EF a; than the two right lines EGF and Ef bound a figure ;
bax. 10. 1. which is impoſſible b; therefore the right line EF is not above,
nor can it be below the plain, for the ſame reaſon; therefore it is
in the ſame plain. Wherefore, &c.
a 3.
.
و
ܪ
PRO P. VII. and VIII. T H E O R.
1
If two right lines be perpendicular to the ſame plain, theſe right
lines are parallel; and, if two right lines are parallel, and one
of them is perpendicular to ſome plain, then the other is perpendi-
cular to the ſame plain.
;
Let two right lines AB, CD, be perpendicular to the ſame
plain, then AB is parallel to CD ; and, if AB be parallel to
CD, and AB be perpendicular to ſome plain, then CD is per-
pendicular to the ſame plain.
Firſt, let AB, CD, be perpendicular to ſome plain, and let
them meet it in the points B, D; join BD; and, in the point D,
draw ED at right angles to BD, and equal to AB ; join BE,
AE, AD; then, becauſe AB is perpendicular to the plain in
which BDE is, it will be at right angles to all the lines drawn in
it, and touching AB^; but AB touches BD, BE, in the ſame
plain ; therefore each of the angles ABD, ABE, is a right
angle. For the ſame reaſon, each of the angles CDB, CDE, is
a right angle; then, becauſe AB is equal to DE, and BD com-
mon, the two lines ED, DB, are equal to AB, BD; and the
angle
a det. 3.
OF EUCLID.
107
>
с
1
d.
C 2.
ܪ
angle ABD equal to the angle EDB ; for each is a right one ; Boo XI.
therefore the baſe EB is equal to the baſe AD b; therefore EB,
BA, are equal to ED, DA; and the baſe AE common; there- b 4. 1.
fore the angles ABE, EDA, are equal"; but EBA is a right c 8. 1.
angle; therefore EDA is likewiſe a right angle; therefore D
is perpendicular to AD, and likewiſe perpendicular to BD, DC;
therefore BD, DA, DC, are in one plain d; but BD, DA, are ds.
in the ſame plain with AB €; therefore AB, DC, are in one
plain ; and the angles ABD, CDB, right ones; therefore AB,
DC, are parallel
Second, If AB and CD are parallel, and AB perpendicular
to ſome plain, CD is perpendicular to the ſame plain. For, the
,
ſame conſtruction remaining, AB is proved at right angles to
BD, BE ; and ED at right angles to DB, DA; and, becauſe
AB is parallel to CD, and DB joins them, CD, AB, are in the
fame plain with DB 6; but ED is proved at right angles to DB, 8 6.
DA ; therefore at right angles to DCh; for DC is in the ſame h 4.
plain with DB, DA ; therefore CD is at right angles to DE,
DBi. Wherefore, &c.
i def. 3.
f 28, I.
9
a
8
ز
PROP. IX.
T H E O R.
R
IGHT lines that are parallel to the ſame right line, altho'
not in the ſame plain with it, are parallel to one onother.
Let the right lines AB, CD, be each parallel to the right
line EF, but not in the ſame plain with it, then AB will be
parallel to CD. For, in EF, affume any point G, and draw
GH at right angles to EF, in the ſame plain paſſing through EF,
AB; and likewiſe GK at right angles to EF, in the plain paſ-
ſing through EF, CD; then, becauſe EF is at right angles to
GÉ, GK, it is alſo at right angles to the plain paſſing through
GH, GK“; but EF is parallel to AB ; therefore AB is alſo at
right angles to the plain paſſing through GH, GKb. For the
{ame reaſon, CD is perpendicular to the ſame plain ; therefore
AB is parallel to CD.; for each is at right angles to the ſame
plain. Wherefore, &c.
a
a 4
b)
D
PRO P. X. THE O R.
IF F two right lines touching one another, be parallel to two other
right lines touching one another, but not in the ſame plain;
theſe right lines contain equal angles,
Let
108
Τ Η Ε Ε L Ε Μ Ε Ν Τ S
Book XI. Let two right lines AB, BC, touching one another, be parąl.
Jel to two right lines DE, EF, touching one another, but not in
the ſame plain, the angle ABC is equal to the angle DEF. For,
take BA, BC, ED, EF, equal to one another, and join AD,
CF, EB, AC, DX; then, becauſe AB, DE, are equal and parallel,
AD, BE, that join them, are likewiſe equal and parallela. For
the ſame reaſon, CF, BE, are equal and parallel; then AD, CF,
are equal and parallel b; therefore AC, DF, that join them, are
equal and parallel ~; then, ſince AB, BC, are equal to DE,
,
EF, and the baſes AC, DT, are equal, the angle ABC is equal
to DEFC, Wherefore, &c.
a 33. I.
be.
a
C 8. I.
PRO P. XI. PRO B.
T
10 let fall a perpendicular on a given plain from a given point
above it.
a I2. 1.
b II. I.
C 31. 1.
Let BH be the given plain, and A the point above it; it is
required from the point A to let fall a perpendicular upon the
given plain BH. In the plain BH take any right line BC; and
from the point A draw AD perpendicular to BC2. If AD is
perpendicular to BH, what was required is done ; if not, draw
DE in the plain at right angles to BC); and from A draw AF
perpendicular to DE а, and through F draw GH parallel to BCS.
Then, becauſe BC is perpendicular both to DA and DE, it is
d def. 3. perpendicular to the plain paſſing through DA, DE d; but GH
is parallel to BC; therefore GH is perpendicular to the plain
palling through DA, DE ®; therefore AF is perpendicular to
GHf; but AF is perpendicular to DE ; therefore at right angles
to the plain paſſing through GH, ED 8; that is, to BH. Where-
fore, &c.
3
d.
C 7.
f def. 3.
& 4.
PRO P. XII. PRO B.
T:
O erect a right line perpendicular to a given plain from a
given point in it.
”
و
Q II
b 31. 1.
It is required to draw a perpendicular to the plain MN, from
a given point A in it. From ſome point B, above the plain, let
fall a perpendicular BC upon ita; and from the point A draw
AD parallel to BC b. Then, becauſe AD, BC, are two paral-
lel right lines, BC, one of them, is perpendicular to the plain
MN, the other, AD, is perpendicular to the same plainº. Where-
fore, &c.
PROP.
676
OF EUCLI U.
109
PROP. XIII. THEOR.
Book XI.
TH
WV O right lines cannot he drawn at right angles to a given
plain, from a point given therein.
a 30
For, if poſſible, let the right lines AB, AC, be drawn perpen-
dicular to a given plain, from the given point A ; let a plain
paſſing through AB, AC, cutting the given plain through A,
in the right line DAE~; but the right line DAE being in the
given plain touches it ; therefore AB, AC, DAE, are in one
plain; then, becauſe AC is perpendicular to the given plain,
the angle CAE is a right angle b; for the ſame reaſon BAÉ is a
right angle; therefore the angle BAE is equal to the angle
CAE, a part to the whole, which is abſurd." Wherefore, &c.
a
a
b def. 3.
PRO P. XIV.
THEOR.
T
HOS E plains to which the ſame right line is perpendicu-
lar, are parallel to each other.
a
а
If the right line AB be perpendicular to each of the plains
DC, EF; then theſe plains are parallel: For, if not, let them
be produced till they meet each other; and let the right line
GH be their common ſection; in which, take any point K,
and join AK, BK'; then, becauſe AB is perpendicular to the
plain EF, it is perpendicular to the right line BK, being in a def. 3,
the ſame plain produced; therefore ABK is a right angle; for
the fame reaſon BAK is a right angle; that is, two angles in a
triangle equal to two right angles, which cannot beb; there-
fore the plains CD, EF, being produced, will not meet each
other; therefore parallel. Wherefore, &c.
PRO P. XV. THE O R.
.
a
b 17,
I F two right lines, touching one another, be parallel to two o-
ther right lines, touching one another, and not in the ſame
plain with them, the plains drawn through theſe right lines are
parallel to each other.
Let AB, BC, two right lines touching one another, be pa- •
rallel to two right lines DE, EF, touching one another, but not
in the ſame plain with them; then the plains paſſing through
AB, BC, DE, EF, being produced, will not meet each other:
For, from the point B, draw the right line BG“, to the point a 11.
a
Gi
LIO
THE ELEMENTS
c def. 3.
C
Book XI. G, in the plain paſſing through DE, EF, and perpendicular to
mit. From the point G, draw GH parallel to DE, and GK pa-
b 31. 1.
rallel to EF); then, becauſe BG is perpendicular to the plain
paſſing through DE, EF, it is perpendicular to all the right lines
touching it in that plain; therefore BG is perpendicular to
GH, GK; and, ſince BA is parallel to GH, the angles GBA,
d 29.1.
BGH, are equal to two right anglesd; but BGH is a right
angle; therefore ABG is likewiſe a right angle: For the ſame
reafon, GBC is a right angle; therefore BG is at right angles
to the plain paffing through BA, BC; but it is likewiſe per-
pendicular to the plain paſſing through DE, EF; therefore BG
is perpendicular to the plains paſſing through BA, BC, and
ED, EF f : Therefore theſe plains are parallel. Wherefore,
&c.
4.
147.
PRO P. XVI. THE O R.
IF
F two parallel plains are cut by another plain, their common
ſections will be parallel.
Let the two parallel plains AB, CD, be cut by any plain
EFGH, whoſe common ſections are EF, GH; then EF is
parallel to GH: For, if EF, GH, are not parallel, if produced,
they will meet, either toward F, H, or E, G. Let them meet
in K: Then, becauſe EFK is in the plain AB, all points taken
in it are in the ſame plain"; therefore K is in the plain AB:
For the ſame reaſon, K is in the plain CD; therefore the plains
AB, CD, meet each other ; but they are parallel ; therefore
they cannot meet ; for the ſame reaſon they cannot meet, if
produced toward E, G; therefore the common ſections EF, GH,
are parallel. Wherefore, &c.
2
7.
ܪ
PRO P. XVII. THE O R.
F two right lines are cut by parallel plains, they will be cui
in the ſame proportion,
>
Let the two right lines AB, CD, be cut by parallel plains
GH, KL, MN, in the points A, E, B, C, F, D ; then AE
will be to EB as CF is to FD: For, let AC, BD, AD, be
joined; and let AD meet the plain KL, in the point X, join
EX, XF; then, becauſe the plains KL, MN, are cut by the
plain
OF EUCLID.
III
2
plain EBDX, their common ſections EX, BD, are parallel“;Book VI.
for the ſame reaſon, the common ſections XF, AC, are paral-
lel; then, fince EX is parallel to BD, AE is to EB as AX is a 16.
to XDb; for the ſame reaſon AX is to XD as CF is to FD ; b 2. 6.
therefore AE is to EB as CF is to FDC. Wherefore, &c.
ܪ
CII.5.
PRO P. XVIII. THE O R.
LE a right line is perpendicular, to fome plain, then all plains
paling through that line will be perpendicular to the fanie
plain.
.
Let the right line AB, be perpendicular to the plain CL ;
then all plains paſſing through AB are perpendicular to the
ſame plain. For, let a plain DE paſs through the right line
AB, whoſe common ſection with the plain ÇL is the right
line CE; from fome point F, in CE, draw FG in the plain DE,
perpendicular to the right line CE ; then, becauſe AB is per-
pendicular to the plain CL, it is perpendicular to the right line
CE in ita; therefore the angle ABF is a right angle; but a def. 3.
GFB is likewiſe a right angle; therefore AB is parallel to
TGb; but AB is at right angles to the plain CL, therefore b 28. 1.
FG is at right angles to the fame plain : For the fame reaſon, c 7.
all other lines drawn perpendicular to the common ſection of
any plain palling through AB, is perpendicular to CL: Then,
becauſe AB, FG, are drawn in one plain, perpendicular to CE,
the common ſection of the plains CH, CL, the plain CH is
perpendicular to the plain CL. Wherefore, &c.
d def. 45
;
C
PRO P. XIX. TH E O R.
I ,
F two plains cutting each other, be perpendicular to ſome plain,
their common ſection will be perpendicular to that ſame plain.
Let two plains AB, BC, cutting each other, be perpendicu.
lar to ſome third plain, ADC; their common ſection BD is
perpendicular to the plain ADC: For, if BD is not perpendi-
cular to ADC, from the point D, draw DE in the plain AB, a def. 4,
at right angles to AD a; and DF in the plain CB, at right
angles to DC; then the two right lines DE, DF, are drawn
from the ſame point, each at right angles to the ſame plain
ADC a ; which is impoſſible b; therefore BD is perpendicular 6 13;
b
to ADC. Wherefore, &c.
.
PROP.
a
b
!
|
II2
Τ Η Ε Ε L Ε Μ Ε Ν Τ S
Book XI.
PRO P. XX. T H E O R.
F a ſolid angle be contained under three plain angles, any two
,
of them, however taken, are greater than the third.
a 23. I.
2
Let the folid angle A, be contained under three plain angles
BAC, CAD, BAD, any two of them, however taken, are
greater than the third.
If the three angles, or any two of them, are equal, then any
two of them muſt be greater than the third ; but, if not equal,
let one of them, as BAC, be the greater: At the point A, make
the angle BAE, with the right line AB, in the plain paſſing
through BA, AC, equal to the angle DAB?; make AE equal
to AD; through E, draw BEC, cutting the right lines AB,
AC, in the points B, C; and join DB, DC; then, becauſe
DA is equal to AE, the two fides BA, AD, are equal to the two
fides BA, AE ; and the angle BAD equal to the angle BAED;
then the baſes BD, BE, are equal"; but the two ſides BD, DC,
are greater than the third BCd; and BD is equal to BE; there-
fore the remainder DC, is greater than EC; and the ſides CA,
AD, are equal to the two fides CA, AE, and the baſe DC
greater than CE; therefore the angle DAC is greater than
EAC; but the angles BAE, EAC, are equal to BAC; there.
;
fore the angles BAD, DAC, are greater than BAC. After the
ſame manner, any other two angles may be proved greater than
the third. Wherefore, &c.
b conſtr.
C4. I.
d 20, I.
е
© 25. I.
a
PRO P. XXI.
ELEN
VERY ſolid angle is contained under plain angles, together
leſs than four right angles.
Let the angle A be a ſolid angle, contained under the plain
angles BĄC, BAD, DAC; theſe angles are leſs than four
right angles. For, in the lines, AB, AD, AC, take any points
B, D, C, and join BD, DC, BC; then, becauſe the folid angle
at B is contained under three plain angles, CBA, ABD, DBC,
any two of which are greater than the third ; the two angles,
a
CBA, ABD, are greater than DBC; for (the ſame reaſon, the
angles BCA, ACD, are greater than BCD; and CDA, ADB,
are greater than BDC: Therefore the fix angles ABD, ABC,
ACB, ACD, ADC, ADB, are greater than the three angles
DBC,
ވެ
am
OF EUCLID.
113
+
с
9
3
C
DBC, BDC, BCD, but theſe three argles are equal to two Book Xi.
right angles; therefore the fix angles ABD, ABC, ACB,
ACD, ADC, ADB, are greater than two right angles : But 32. 1.
the three angles of every triangle are equal to two right angles;
therefore the nine angles CBA, BCA, BAC, ACD, CAD,
ADC, ADB, ABD, DAB, are equal to fix right angles : But
fix of which are proved greater than two right angles; there-
fore the remaining three angles BAC, DAC, BAD, which
contain the ſolid angle A, are leſs than four right angles : 112
the ſame manner, it may be proved, if the angle is contained by
more than three plain angles, that theſe are together leſs than
four right angles. Wherefore, &c.
PRO P. XXII. T H E O Pe.
IF F there be three plain angles, whereof any two, however taken,
are greater than the third, and the right lines that contain
them be equal, then it is poſible to make a triangle of the right
lines, joining the equal right lines, which form the angle.
ܪ܂
9
و
9
ز
)
Let ABC, DEF, GHK, be three given plain angles, any two of
which are greater than the third ; and let AE, BC, ED, EF, GH,
HK, be the equal right lines that contain them; and join AC,
'DF, GK ; then, of, theſe three right lines, a triangle may be
made. For, if the angles B, E, H, or any two of them, are equal,
then any two of them muſt be greater than the third a, and a 4. 1.
likewiſe their baſes ?; of which, let AC be greater than DF, or
a
GK; then DF and GK are greater than AC. For, make the
angle ABL equal to the angle GHK b, and make BL equal to b 23. fi
either AB, BC, DE, EF, GH, HK; and join AL, CL;
then the two fides AB, BL, are equal to the two ſides GH,
HK, each to each; and they contain equal angles; therefore
the baſe AL will be equal to the baſe GK °; and, fince the
angles at E and H are greater than the angle ABC, the angle
GHK is equal to the angle ABL ; therefore the angle ar E is
greater than LBC“; but the two fides LB, BC, are equal to c hyp.
DE, EF, each to each; and the angle DEF greater than LBC;
then the baſe DF is greater than LC4; but GK is proved equald 24. 1.
to AL; therefore DF, CK, are greater than Al, LC; but
AL, LC, are greater than AC °; therefore DF, GK, are much e 20.1.
greater than AC; therefore any țwo of the right lines AC, DF,
GK, are greater than the third: Therefore a triangle may be
made, whoſe fides are equal to the three given right lines.
Wherefore, &c.
PROP.
;
i
}
P
114
THE ELEMENTS
Book XI.
PRO P. XXIII. PRO B.
Tº make a ſolid angle of three plain angles, any two of which
are greater than the third ; but theſe three angles muſt, toge-
ther, be leſs than four right angles.
2 22.
d 22. I.
C 5.4
ز
d 8. I.
s. 1.
e.
f hyp.
It is required to make a ſolid angle of three plain angles ABC,
DEF, GHK, any two of which are greater than the third ; and
all the angles together leſs than four right angles. Let the right
lines AB, BC, DE, EF, GH, HK, be made equal to one ano.
ther; and join AC, DF, GK. Then a triangle may be made a of
three right lines equal to AC, DF, GK; which let be LMN;
make the fide LM equal to AC, MN to DF, and LN to
GKb; deſcribe the circle LMN about the triangle"; its center,
X, is either within the triangle, upon one of the fides, or with-
out the triangle. Firſt, let it be within the triangle, and join
LX, MX, NX; then, if AB be not greater than LX, it will
be either equal or leſs. Firſt, let it be equal; then AB, BC,
are equal to LX, XM, and the baſe LM equal to AC ; then the
angle LXM is equal to the angle ABC d. For the ſame reaſon,
the angle MXN is equal to DEF, and NXL to GHK; but the
three angles LXM, MXN, NXL, are equal to four right
angles €; therefore ABC, DEF, GHK, are equal to four right
angles; but they are leſs f; which is abſurd; therefore LX, XM,
are not equal to AB, BC; and they are not greater. For, if
poſſible, let LX, XM, be greater than AB, BC, and cut off
XO, XP, equal to AB, BC; join OP. Then, becauſe AB is e-
qual to BC, and XO to XP, the remainders LO, MP, will be
equal ; therefore OP is parallel to LM $; and the triangles LXM,
OXP, are equiangular h; therefore XO is to OP as XL is to
LMj; and, by altern. XO is to XL as OP is to LM. But LX is
greater than X0; therefore LM is greater than OP. But LM
is put equal to AC; therefore AC is greater than OP, therefore
the angle ABC will be greater than the angle OXPk. For the
ſame reaſon, DEF is greater than MXN, and GHK than NXL;
but OXP, WXN, NXL, are equal to four right angles"; there-
fore the angles ABC, DEF, GHK, are greater than four right
angles, and likewiſe leſs f; which is impoſſible : Therefore
LX, XM, are not greater than AB, BC; but they are proved
not equal; therefore they are leſs; therefore, on the point X,
raiſe XR perpendicular to the plain of the circle LMNI, and e-
qual to the exceſs by which the ſquare of AB exceeds the ſquare
of LX; and join RL, RM, RN. Then, becauſe RX is perpen-
dicular to the plain LMN, it is at right angles to LX, MX,
NX; therefore the ſquares of LX, XR, are equal to the ſquare
of
8 2. 6.
h 29. 1.
i 4. 6.
و
.
k 25. 1.
1
>
I 12+
m def. Go
OF EUCLID.
IIS
;
و
,
ز
of LR". For the 'ſame reaſon, the ſquares of RX, XM, are e-Book VI.
qual to the ſquare of RM; and the ſquares of RX, XN, equal
to the ſquare of RN; but LX, MX, NX, are equal, and RX n 47. I.
common; therefore LR, RM, RN, are equal; but the ſquare
of AB is equal to the ſquares of LX, XR0; therefore LR is e. o conft.
qual to AB. But BC, ED, EF, GH, HK, are each equal to
AB; therefore RL, RM, RN, are each equal to AB or BC;
and the baſe ML equal to AC; therefore the angle MRL is e-
qual to the angle ABC; but MR, RN, are equal to DE, EF,
and the baſe MN to DF, the angle MPN to DEF, and the
angle LRN to GHK; therefore the ſolid angle at R is contained
by the three plain angles LRM, MRN, LRN, equal to the
three plain angles ABC, DEF, GHK.
Now, let the center of the circle X be on one ſide of the tri-
angle, viz. MN; join XL; then AB is greater than LX. For,
if not, it will be either equal or leſs. Firſt, let AB be equal to
LX; then MX, XL, are equal to AB, BC; that is, MX, XL,
are equal to MN; but MN is equal to DF; therefore DE, EF,
are equal to DF; which is abfurd P; much leſs can MX, XL, p 20. I.
that is, MN, that is, DF, be greater than DE, EF; therefore
AB is greater than LX; and, if XR is drawn perpendicular to
the plain LMN, and equal to the exceſs by which the ſquare of
AB exceeds the ſquare of LX, the figure can be conſtructed as
before.
Laſtly, let the center X of the circle be without the triangle
LMN; join LX, MX, NX; then AB is greater than LX. If
not, it is either equal or leſs. Firſt, let it be equal; then the
two ſides AB, BC, are equal to the two fides MX, XL; and the
baſe AC equal to ML ; therefore the angle ABC is equal to
the angle MXL d. For the ſame reaſon, GHK is equal to d 8. 1.
d
LXN, but the whole angle MXN is equal to the angles MXL,
NXL; therefore MXN is equal to ABC, GHK; that is, DEF
is equal to ABC, GHK; but ABC, GHK, is greater than
DEF", and likewiſe equal; which is abſurd ; therefore AB is 9 20.
;
not equal to LX. Let AB be leſs than LX, and make OX,
XP, equal to AB, BC, then the remainders OL, MP, will be
equal; therefore OP is parallel to ML, and the triangles equi- : 2. G.
angular ; therefore XO is to OP as XL is to LM ; by altern.
as XO is to XL, fo is OP to LM; but XL is greater than XO;
therefore LM is greater than OP; but LMis equal to AC;there-
fore AC is greater than OP, and the angle ABC greater than
OXPk. Draw XV equal to XO or XP ; and join OV; then thek 25. 1.
angle GHK is greater than OXV. At the point X, with the right
line LX, make the angle LXS equal to ABC; and the angle
LXT to GHK"; and XS, XT, cach equal to XO; and join r 2 je 5.
OS, OT, ST; then, becauſe the two ſides AB, BC, are equal
ز
;
a
ܪ
1
to
116
T H E E L EM ENTS
S.
S
S 4. I.
t
t 24. I.
Book XI. to the two fides OX, XS, and the angle ABC to OXS; the
baſe AC, that is, LM, will be equal to OS . For the ſame
reaſon, LN is equal to OT; and, ſince the two fides ML, LN,
are equal to the two fides OS, OT, and the angle MLN, or
POV, greater than SOT, for it contains it, the baſe MN is
greater than ST"; but MN is equal to DF; therefore LF is
greater than ST; therefore the angle DEF is greater than
SYTk; but the angle SXT is equal to the angles ABC, GHK;
therefore the angle DEF is greater than ABC, GHK, and
likewiſe lefs; which cannot be; therefore AB is not leſs than
LY; but it has been proved not equal to it; therefore muſt
be greater. Then, make XR equal to the exceſs by which the
{quare ot" AB exceeds the ſquare of LX; and join RM, RN,
RL; then, in the fame manner, it may be proved, that the ſo-
lid angle R is the angle required. Wherefore, &c.
k 25. I.
.
1
9
PRO P. XXIV. T H E O R.
berasal
I
F a ſolid be contained by ? x parallel plains, the optoſite plains
thereof are equal parallelograms.
a 16.
>
b 23. I.
Let the folid CDGH be contained by the parallel plains AC,
GF, BG, CE, FP, AE, the oppofte plains thereof are equal
parallelograms. For, becauſe the parallel plains BG, CE, are
cut by the plain AC, their common ſections AB, CD, are pa-
raliela; and, becauſe the parallcl plains BF, AE, are cut by the
plain AC, their common ſections AD, BC, are parallel ; there-
fore AC is a parallelogram. In the ſame manner, it is proved
that GF is a parallelogram. Then, becauſe BH, AG, CF, DE,
join the parallel lines AD, GE, BC, HF, they are equal to
one another h. For the ſame reaſon, AB, HG, CD), EF, are e-
qual to one another ; therefore BG, CE, AC, GF, AE, BF,
are parallelograms. Join AH, DF; then, becauſe AB, BH,
are parallel to DC, CF, the angle ABH is equal to DCF;
then, becauſe AB, BH, are equal 10 DC, CF, and the angles
ADH, DCF, equal, the baſes AH, DF, are equal d; but the
parallelogram BG is double the triangle ABH, and CE double
CDF"; therefore the parallelograms BG, CE, are equal; in the
fame manner, the parallelograms AC, GI, are proved equal;
and AE equal to BF. Wherefore, &c.
2
CIO,
d 4, I,
.
e 34. I.
e
--
PROP.
L
OF EUCLI V.
117
t
Book XI.
PRO P. XXV. THEO R.
IE a ſolid parallelopipedon be cut by a plain parallel to oppoſite
plains; then, as baſe is to baſe, jo is folid to folid.
>
و
Let the folid ABCD be cut by a plain YE, parallel to the op-
poſite plains RA, DH; then, as the baſe EFU A is to the baſe
EHCF, fo is the ſolid ABFY to the folid EGCD. For, pro-
duce AH both ways, and make AK, KL, each equal to AE ;
and HM, MN, each equal to EH ; and compleat the parallelo-
grams LO, KU, HX, MS, and the ſolids LP, KR, HQ,
MT, then, becauſe the right lines LK, KA, AE, are equal,
the parallelograms LO, KU, AF, are equal“; as alſo the pa. a 36. 1.
rallelograms KV, KB, AG, and the parallelograms LW, KP,
AR. For the ſame reafon, the parallelograms EC, HX, MS,
are equal; as alſo, HG, HI, IN, and the parallelograms DH,
MQ, NT. Then, becauſe LK, KA, AE, are equal, and like-
mife HM, MN, each equal to HE; LE is the ſame multiple of
AE that LF is of AF; and EN the ſame multiple of EH that
ES is of EC ; and LG of AG; LR of AR; ET of HG; and
;
EQ, of EY. Wherefore the three plains in the folid LP, and
the three oppoſite ones, which are equal to them, are equal to
the three plains in the folid KR, or AY, and the three op-
ofite plains which are equal to them b; therefore the three ſolids b 24.
LP, KR, AY, are equals, and the ſame multiple of AY that c def. 10.
LF is of AF. For the ſame reaſon, the ſolids ED, HQ, MT,
are equal; therefore ET is the ſame multiple of ED that ES is
of EC: Wherefore, if LF be equal to ES, the ſolid LY will
be equal to the folid NY, if greater, greater, and, if leis, leſs ;
Wherefore, as AF is to FH, ſo is the folid AY LO ED d. Where. d def. 5. 5,
fore, &c.
с
PRO P. XXVI. PROB.
А.
I a given point, in a given right line, to make a ſolid angla
cqual to a ſolid angle given.
It is required, at a given point A, in a given right line AB,
to make a ſolid angle equal to the ſolid angle contained by the
plain angles EDC, EDF, FDC. In the right line DF affume
any point F; from which draw FG perpendicular to the plain
pailing through ED, DC, meeting the plain in the point
G;
+
j18
THE ELEMENTS
a II.
C 12.
Book XI. G^; join DG; at the point A, with the right line AB, make
w the angles BAL, BAK, equal to the angles EDC, EDG b; and
make AK equal to DG; at the point K, in the plain BAL,
h 23. I. raiſe a perpendicular HK C: which make equal to GF; and join
HA; then the ſolid angle at A, which is contained by the
plain angles BAL, BAH, HAL, is equal to the ſolid angle
at D, contained by the plain angles EDC, LDF, FDC. For,
take the right line AB, equal to DE; AL to DC; and join
HB, KB, FE, GC, FC; then, becauſe GF is perpendicular
def. 3.
to the plain EDC 4, the angles FGD, FGE, FGC, are right
angles; for the ſame reaſon, HKA, HKB, HKL, are right
angles; and, becauſe the two ſides KA, AB, are equal to the
two ſides GD, DE, and contain equal angles, the baſes BK,
EG, are equal ; and, becauſe BK, KH, are equal to EG, GF,
each to each, and contain equal angles, the bafes HB, FE, are
equale. Again, becauſe AK, KH, are equal to DG, GF, each
to each, and contain equal angles, the bale AH is equal to DF;
but AB, AH, are equal to DE, DF, and the baſe BH equal to EF;
8, lo
therefore the angle BAH is equal to EDF f; but the angle BAL
is equal to EDC, and a part BAK equal to EDG; therefore
the remainders KAL, GDC, are equal, and the baſe KL to
GC“; and, becauſe HK, KL, are equal to FG, GC, each to
each, and the angle HKL equal to FGC, the baſe HL is equal
to FC ; but HA, AL, are equal to FD, DC, and the baſe
HL equal to FC; the angle HAL is equal to FDC; therefore
the plain angles BAL, BAH, HAL, containing the ſolid angle
A, are equal to the plain angles EDC, EDF, FDC, containing
the folid angle at D, each to each ; therefore the ſolid angle at
def. 1o. A is made equal to the ſolid angle at Ds; which was to be
done.
4. 1.
1
PRO P. XXVII. PRO B.
T9
o deſcribe a parallelopipedon from a given right line, ſimilar
and alike ſituate to a ſolid parallelopipedon given.
It is required to deſcribe, from the right line AB, a ſolid pa-
rallelopipedon, ſimilar and alike ſituate to the given ſolid paral-
lelopipedon CD.
At the point A, in the given right line AB, make a ſolid
angle A, contained by the plain angles BAH, HAK, KAB, e-
qual to the ſolid angle at Ca, ſo that the angle BAH be equal
to ECF; BAK to ECG; and HAK to FCG; and make B A to
AK
26.
OF EU CLID.
119
AK as EC is to CG 6; and KA to AH as GC is to CF; then“, Book XL-.
by equality, as BA is to AH, ſo is CE to CF. Compleat the
parallelogram BH, and ſolid AL: Then, becauſe the three plain b 12. 6.
angles, containing the ſolid angle at A, are equal to the three plain C 22. so
angles containing the folid angle at C, and the ſides about the
equal angles proportional, the parallelogram KB is fimilar to
the parallelogram GE. For the ſame reaſon, KH is ſimilar to
GF, and HB to FE ; therefore the three parallelograms of the
folid AL are fimilar to the three parallelograms of the folid CD;
but theſe three parallelograms are equal and ſimilar to the three
oppoſite ones d; therefore the folid AL is ſimilar to the folid
CD 4: Which was to be done.
& 24.
e def. De
PRO P. XXVIII. T H E O R.
IF
a folid parallelopipedon be cut by a plain paſſing through the di-
agonals of two oppoſite plains, that ſolid will be biſected by the
plain.
1
>
If the folid parallelopipedon AB be cut by the plain GAEF,
paſling through the diagonals GF, AE, of two oppoſite plains,
then the folid AB is bifected by the plain GAEF. For, becauſe
the triangles CGF, GBF, are equal, and likewiſe the triangles
ADE, A EH", and the parallelograms AC, BE , for they are a 34. I.
oppoſite, and likewiſe GH equal to CE ; the priſm contained
by the two triangles CGF, ADE; and the three parallelograms
GE, AC, CE, is equal to the priſm contained by the triangles
GFB, AEH, and the three parallelograms GË, BE, AB “.
Wherefore, &c.
b 외동​이
​c def. Io,
с
PRO P. XXIX. THEO R.
<
с
OLID parallelepipedons, conſtitute upon the ſame baſe, ha-
ving the ſame altitude, and whoſe inſiſtent right lines are in
the ſame right line, are equal to one another.
S
Let the ſolid parallelopipedons CM, BF, be conſtitute upon
the ſame baſe AB, having the ſame altitude, and whoſe inſiſtent
right lines AF, AG, LM, LN, CD, CE, BH, BK, are in
the ſame right lines FN, DK ; then the ſolid CM is equal to
the folid CŇ. For, becauſe CH, CK, are parallelograms,
DH, EK, are each equal to CB ; therefore DH is equal to a 34. I.
EK. Take EH from, or add to both, then there will remain.
HK
I20
THE ELEMENTS
b 8. I.
C 24.
>
Book XI. HK equal to DE, and the triangle DEC equal to HKB b, and
Wthe parallelogram DG to HN; but the parallelogram CF is e-
qual to BM', and CG to BN, for they are oppoſite; therefore
the priſm contained by the two triangles AFG, DEC, and the
three parallelograms CF, DG, CG, is equal to the priſm con-
tained by the two triangles LMN, HKB, and the three paralle.
d def. 10. lograms BM, HN, BNå; add, or take away the folid whoſe baſe
is the parallelogram AB, oppoſite to the parallelogram GEHM;
then the folid ČM is equal to the folid CN. Wherefore, &c.
d
i
PRO P. XXX.
T H E O R.
OLID parallelopipedons, conſtitute upon the ſame baſe, ha-
ving the ſame altitude, and whoſe infijtent right lines are not
in the ſame right line, are equal to one another.
S
>
1
a 24
b
3
b 29.
Let there be folid parallelopipedons CM, CN, having equal
altitudes, ſtanding on the ſame baſe AB, and whoſe inſiſtent
right lines AF, AG, LM, LN, CD, CE, BH, BK, are not
in the ſame right lines; then the
ſolid CM will be equal to the
ſolid CN. For, produce NK, DH, till they meet in R; and
draw GE, FM, meeting in X; likewiſe produce GE, FM, to
the points 0, P; join AX, LO, CP, BR; then the folid CM,
whoſe baſe is the parallelogram ACBL, oppoſite to the equal
parallelogram FDHM“, is equal to the ſolid CO 6, whoſe baſe
is the ſame parallelogram AB, oppoſite to the equal parallelo-
gram XR; for they ſtand upon the ſame baſe AB, and the in-
;
liſtent lines AF, AX, LM, LO, CD, CP, BH, BR, are in
the ſame right lines FO, DR; but the ſolid CO is equal to the
folid CN, for they have the fame baſe AB, oppoſite to the
parallelograms XR, GK, each equal to AB, and their infift-
ent right lines AG, AX, CE, CÔ, LN, LO, BK, BR, are in
the ſame right lines GP, NR; therefore the folid CM is equal
to the ſolid CN. Wherefore, &c.
>
PROP. XXXI. THEOR.
·
1
}
S
OLID parallelopipedons, conſtitute upon equal baſes, and
having the ſame altitudes, are equal.
Let AE, CF, be folid parallelopipedons, conſtitute upon the e-
qual baſes AB, CD; and having the ſame altitude the folid AE is
equal
1
1
OF EU CLI D.
I 21
ز
ލް
d
,
و
d def. Io
equal to the ſolid CF. Firſt, let the folids AE, CF, have the Book XI.
, .
infiftent lines AG, HK, LM, BE, OP, DF, CG, RS, at
right angles to the baſes AB, CD; and let the angle ALB be
equal to the angle CRD. Produce CR to T; and make RT e-
qual to LB; compleat the parallelogram DT, equiangular to
AB or CD; and the ſolid KI, having its inſiſtent right lines
at right angles to DT, and of the fame altitude with AE or CF.
Then, becauſe the right lines DF, RS, are at right angles to
the plain OT, they are parallela and equal b; therefore the a 6.
,
b const.
parallelogram DS is equal and parallel to CP, TÍ: Therefore
the ſolid CF is to the folid Pil as the baſe OR is to the baſe DT ; C 250
but OR is equal to DT ; therefore the ſolid CF is equal to the
folid RI. But the folid RI is equal to the folid AE; therefore
the ſolid AE is equal to the folid CF: But, if the angle ALB
is not equal to CRD, at the point R, with the right line RF,
make the angle TRY equal to the angle ALB ; and make RY
equal to AL ; and compleat the parallelogram RX, and folid
YW. Produce DR, VI, XY, to the points Q and a ; and
compleat the ſolid ae ; then the parallelograms RX, RQ, are
equald; and, becauſe RX is equiangular to .:B, and the inſiſt- d 35. I.
ent lines at right angles to the bale RX, and of the ſame alti-
tude with the folid AL, the plains in the folid AE are equal and
ſimilar to theſe in the ſolid YW; therefore the ſolid YW is en
qual to the ſolid AEd. For the ſame reaſon, the folid aW,
whoſe baſe is the parallelogram RW, and ae, that oppoſite to
it, is equal to the ſolid YW, whoſe baſe is the parallelogram
RW, and Yf, that oppoſite to it; for they ſtand upon the
ſame bale RW, have the ſame altitude, and their inſiſtent lines
Ra, RY, TX, TQ, SZ, SN, We, Wf, are in the ſame right
lines aX, Zf; but the folid YW is equal to the ſolid CF;
therefore the ſolid aW is equal to the ſolid CF.
Now, let the inſiſtent lines ML, EB, GA, KH, NO, SD,
PC, FR, not be at right angles to the baſes AB, CD, the ſolid
AE will be equal to the folid CF. For, from the points G, K,
E, M, P, F, S, N, let fall the right lines Mf, ET, GY, Kg,
PX, FW, Na, SI, perpendicular to the plain of the baſes AB,
CD, meeting them in the points f, Y, S, T, X, W, I, a ;e ir
and join fY, Yg, gT, Tf, Xa, XW, WI, la ; then, becauſe
GY, Kg, are at right angles to the fame plain, they are paral.
lelf. For the ſame reaſon, Mf is parallel to ET. But MG isf 6. 6.
.
parallel to EK; therefore the plains MY, KT, of which the
one pafſes through GY, Yf, and the other through Kg, gt,
which are parallel to GY, YF, and not in the ſame plain with
them, are parallel to one anothers, and equal and parallel to their
oppoſite plains ; therefore fE is a parallelopipedon. It may be
proved in the ſame manner, that aF is a parallelopipedon; but
Q
the
a
>
>
e
.
8 IS
>
I 22
THE ELEMENTS
Book XI. the ſolid GT is equal to the ſolid PI; for they are upon equal
baſes, and of the ſame altitude, from what has been demonſtra-
h 29. or 30.
ted; and the ſolid GT is equal to the folid AEh; and the folid
XF to the ſolid aS; therefore the ſolid AE is equal to the folid
CF. Wherefore, &c.
PRO P. XXXII. THEO R.
SOLID parallelopipedons that have the fame altitude are to one
another as their baſes.
;
Let AB, CD, be ſolid parallelopipedons, having the fame al-
titude as the baſe AE is to the baſe CF, ſo is the ſolid AB to
the ſolid CD.
For, to the right line FG, apply the parallelogram FH,
equal to the parallelogram AË; upon the baſe FH, compleat
the ſolid GK, of the ſame altitude as CD; then the ſolid AB is
equal to the ſolid GK a; but the folid CK is cut by the plain
DG, parallel to the oppoſite plain ; therefore the ſolid CD is to
the ſolid GK, as CF is to FHb; that is, AB is to CD as AE is to
CF. Wherefore, &c.
a 31.
b 25.
PRO P. XXXIII. THE O R.
SIMILAR ſolid parallelopipedons are to one' another in the
triplicate ratio of their homologous ſides.
Let AB, CD be ſimilar folid parallelopipedons, and let the
fide AE be homologous to the ſide CF; then the ſolid AB has to
i
the ſolid CD a triplicate ratio of that which the ſide AE has to
CF.
For, produce AE, GE, HE, to K, L, M; make EK equal
to CF, EL to FN, and EM to FR; and the angle KEL is e-
qual to CFN; for AEG is equal to CFN; and compleat the
parallelogram KL, and the ſolid KO; then the parallelogram
KL is ſimilar and equal to the parallelogram CN. For the
ſame reaſon, the parallelogram KM is equal and ſimilar to the
parallelogram CR, and OE to FD; therefore the whole ſolid
KO is equal and ſimilar to the folid CD. Likewiſe, compleat
the parallelogram HL, and folids EX, LP, upon the baſes GK,
KL, having the fame altitude as AB, for EH is an inſiſtent line
to both; but the folid OK is proved ſimilar to CD, and AB is
given
á 24
ز
OF EUCLI D.
123
.
d.
given ſimilar to CD; therefore AB is fimilar to OK b. Then, Book XI.
becauſe AB is ſimilar to CD, and the baſe AG to CN, as AE
is to CF, ſo is EG to FN; and fo is EH to FR ~; and, becauſe b 21. 6.
;
FC is equal to EK, and FN to EL, and FR to EM, as AE is to and 12. s.
EK, fo is EG to EL ; and ſo is the parallelogram AG to GK;
but, as GE is to EL, ſo is the parallelogram GK to KL“; and,
as HE is to EM, fo is the parallelogram PE to KM“; therefore,
as AG is to GK, ſo is GK to KL ; and ſo is PE to KM; but,
as AG is to GK, ſo is the folid AB to the ſolid EX d, for the d. 25.
plain GH is parallel to the oppoſite plains; and, as GK is
to KL, ſo is the folid EX to PL"; and, as PE is to KM,
fo is the ſolid PL to the folid K'o.
Then, becauſe the
four folids AB, EX, P , KO, are proportionals, AB has to
KO a triplicate proportion of what AB has to EX ®; but, as AB e def, 18.
is to EX, ſo is the parallelogram AG to GK d; and ſo is the
right line AE to the right line EK“; therefore the ſolid AB has
to the ſolid KO a triplicate ratio of what AE has to EK ; but
the ſolid KO is equal and ſimilar to the ſolid CD ; and the right
line EK equal to CF : Therefore the ſolid AB has to the folid
CD a triplicate ratio of what the homologous fide AE has to the
homologous fide CF. Wherefore, &c.
Cor. Hence, if four right lines be proportional, as the firſt
is to the fourth, fo is a ſolid parallelopipedon deſcribed on the
firſt, to a ſimilar one deſcribed on the ſecond,
e
:
а.
a
PRO P. XXXIV. THE O R.
O
TH
HE baſes and altitudes of equal ſolid parallelopipedons are
reciprocally proportional ; and parallelopipedons, whoſe baſes
and altitudes are reciprocally proportional, are equal.
;
Let AB, CD, be equal ſolid parallelopipedons ; then the
baſe EH is to the baſe NP, as the altitude of the folid CD is to
the altitude of the ſolid AB.
Firſt, let the inſiſtent right lines AG, EF, LB. HK, CM,
NX, OD, PR, be at right angles to their baſes; then, as the
baſe EH is to the baſe NP, ſo is CM to AG. For, if the baſe
EH is equal to the baſe NP; then the altitudes CM, AG, are
equal. For, if the baſes EH, NP, are equal, but the altitudes
not equal, then the ſolids are not equal“; but the ſolid AB is a 31,
put equal to the folid CD; therefore the altitudes CM, AG, are
equal : Therefore, as the baſe EH is to the baſe NP, fo is CM
to AG 6. Now, let the baſes be unequal, and let EH be the 32.
greater, then the altitude CM will be greater than the altitude
AG;
-
2
I24
THE ELEMENTS
C 7. 5.
d 32.
d
3%.
و
>
2
Book XI. G. for the folids are equa
Make CT equal to AG; and up-
on PN compleát the ſolid CV, whoſe altitude is CT. Then, be-
cauſe the ſolids AB, CD, are equal, and CV is ſome other folid,
AB is to CV as CD is to CVC; but AB is to CV as the baſe
EH is to the baſe NP d; and, as the ſolid CD is to CV, ſo is the
e 1. 6. and baſe MP to TP, and ſo is MC to TC ®; therefore the baſe EH
is to NP as MC is to CT ; but CT is equal to AG; therefore
EH is to NP as MC is to AG. And, if the baſes and altitudes
are reciprocally proportional, then AB is equal to CD. For, a
gain, let the inſiſtent right lines be at right angles to the baſes;
then, if the baſes are equal, becauſe the altitudes are as their
baſes, the altitudes are equal; therefore the folids AB, CD?,
are equal. But, if the baſes are not equal, let EH be greater
than NP, then the altitude of the folid CD is greater than the
altitude of the ſolid A5 ; that is, CM is greater than AG.
Put CT equal to AG, and compleat the ſolid CV; then, becauſe
;
the baſe EH is to the baſe NP, as MC is to AG; and Clis e-
I g. $ qual to AG; and, as EH is to NP, ſo is MC to CDf; but, as
MC is to CT, ſo is MP to PT"; and, as EH is to NP, ſo is
AB to CV d, but, as AB is to CV, fo is CD to CV; therefore
AB is equal to CD f.
Secondly, If the inſiſtent right lines FE, BL, GA, KH,
XN, DO, MC, RP, are not at right angles to the baſes; from
the points F, G,.B, K, X, M, D, R, Jet be drawn perpen-
diculars meeting the plain of the baſes EH, NP, in the points
S, T, V, Y, Q, Z, a, f, and compleat the ſolids FV, Xa. Then,
if the ſolids be equal, their baies and altitudes are reciprocal-
ly proportional, viz. as EH is to NP, fo is the altitude of CD to
the altitude of AHF, For, becauſe the folids AB, CD, are equal,
and the ſolid AB is equal to the colid BT, and the ſolid CD to the
29. OP 30.
folid DZ ; therefore the folid BT is equal to the ſolid DZ; but
the baſes and altitudes of equal folids, whoſe inſiſtent right lines
are at right angles to their baſes, are proved to be reciprocally
proportional. Therefore, as the baſe KF is to the baſe Rx, ſo
is the altitude of the ſolid DZ to the altitude of the folid BT
but the ſolids DZ, DC, have the ſame altitude; and the ſolids
BT, BA, have the ſame altitude; therefore, the baſe EH is to
the baſe NP, as the altitude of the ſolid DC is to the altitude of
the folid AB;
Again, let the baſes and altitudes of the folid parallelopipe-
dons be reciprocally proportional, viz. as EH is to NP, ſo let
the altitude of CD be to the altitude of AB ; then the ſolids AB,
CD, are equal. For, the fame conſtruction remaining, as the
baſe EH is to the baſe NP, ſo is the altitude of CD to the alti-
tude of AB; but the altitudes of the ſolids AB, BT, are the
fame, and likewiſe of the folids CD, DZ; therefore, as the
baſe
9
>
8
+
OF EUCLID.
125
7
•
ز
baſe FK is to the baſe XR, fo is the altitude of the folid DZ to Book XI.
the altitude of the folid BT, therefore the baſe and altitudes of
the folid parallelopipedons BT, DZ, are reciprocally propor-
tional: But theſe folid parallelopipedons whoſe inſiſtent right
lines are at right angles to their baſes, and their baſes and alti-
tujes are reciprocally proportional, are equal to each other ; but
the ſolid BT is equal to BA, for they ſtand upon the fame bate
FK, and have the ſaine altitude. For the ſame reaſon, the ſolid
DZ is equal to the folid DC; therefore the folid AB is equal to
the ſolid CD: Wherefore, ſolid parallelopipedons, whoſe baſes
and altitudes are reciprocally proportional, are equal. If the
baſes are not equal, and the innitent right lines not at right
angles to the baſes, the bales and altitudes may be proved reci-
procally proportional, in the ſame manner as when the inſiſtent
right lines are at right angles to their baſes; and in the ſame
manner, if the bafes and altitudes are reciprocally proportional,
the ſolids are equal. Wherefore, &c,
PRO P. XXXV. THE OR
IF
F there be two plain angles equal, and from their vertices two
,
right lines be elevated above the plains in which the angles are,
making equal angles with the lines containing the given angles ;
and iſ, in the elevated lines, airy points be taken, from which per-
pendiculars are let fall to the plains
palling through the given right
lines; then theſe elevated lines Mball be equally inclined to the gi-
von plain.
{
و
a
Let BAC, EDF, be two right lined plain angles, from whoſe
vertices A, D, let two right lines AG, DM, be elevated above
the plains of the ſaid angles, making the angles MDE, MDF,
equal to the angles GAB, GAC, each to each ; then the angle
MDN will be equal to GAL.
For, if AG is not equal to DM, take AH equal to DM ; and
from H draw HK parallel to GL;but GL is perpendicular to the
plain paſlirg thro’BAC; therefore HK is likewiſe perpendicular
to the ſame plain a. From the points K, N, draw the right lines a 7.
KB, KC, NE, NF, perpendicular to the right lines AB, AC,
DE, DF. Join HC, CB, MF, EF, HB, EM; then the ſquare
of HA is equal to the ſquares of HK, KAb, and the ſquare of b 47. .
KA equal to the ſquares of KC, CA; therefore the ſquare of
AH is equal to the ſquares of 'K, KC, CA; but the ſquares
of HK, KC, are equal to the
ſquare of HC; for the angle HKC
is a right one: Therefore the ſquare of HA is equal to the ſquares
of HC, CA; therefore the angles HCA, HBA, are right
6
a
ones.
126
Τ Η Ε Ε L Ε Μ Ε Ν Τ S
20. I.
و
a
Book XI. ones. For the ſame reaſon, the angles DFM, DEM, are
right angles : Therefore the angle AC# is equal to DFM; but
C 48. 1.
the angle HAC & is equal to MDF; therefore the two triangles
d hyp.
HAC, MDF, have two angles, HAC, HCA, in the one, equal
to MFD, MDF, of the other, each to each, and HA, a fide of
the one, equal to MD, a fide of the other; therefore, the re-
maining fides MF, FD, are equal to HC, CA, each to each º.
In the ſame manner, AB is proved equal to DE, and BH to
EM: Therefore, the two fides CA, AB, are equal to the two
fides FD, DE, and the angle BAC equal to FDE ; therefore the
baſe BC is equal to EF, the angle ACB to DFE, and ABC to
DEF, But the angles ACK, DFN, are equal ; for each is a
right one; therefore the remaining angle BCK is equal to EFN.
For the ſame reaſon, the angle CBK is equal to FEN. And,
becauſe the two triangles BCK, EFN, have the two angles
CBK, BCK, in the one, equal to the two angles FEN, EFN,
each to each, and a fide BC equal to EF; therefore the fide FN
is equal to CK ¢ But AC is equal to DF; therefore the two
fides AC, CK, are equal to the two ſides DF, FN, and they
contain right angles; therefore the baſe AK is equal to DN.
And, ſince AH is equal to DM, their ſquares are equal; but
the ſquares of AK, KH, are equal to the ſquare of AH, and the
ſquares of DN, NM, equal to the ſquare of DM; for the angles
AKH, DNM, are right angles; therefore the ſquares of AK,
KH, are equal to the ſquares of DN, NM, and the ſquares of
AK, DN, equal ; therefore the ſquares of KH, MN, are equal;
that is, the right line HK equal to MN; therefore the angle
6.1.
HAK is equal to MDNF, that is, GAL equal to MDN :
Which was to be demonſtrated.
Cor. Hence, if two right lined plain angles be equal, from
whoſe point equal right lines are elevated on the plain of the
angles containing equal angles with the given lines, each to
each ; perpendiculars drawn from the extreme points of theſe
elevated lines to the plain of the given angles, are equal to one
another.
ܪ
>
i
PROP. XXXVI.
THEOR.
F
If three right lines A, B, C, are proportional, the ſolid parallelo-
pipedon made of them is equal to the ſolid parallelopipedon made
of the middle line, it being equiangular to the former parallelo-
pipedon.
Let
OF EU C 127
CL
LIV.
ލް
a
a
Let the three proportional right lines be A, B, C, viz. A to Book XI:
B as B is to C; then the ſolid made with A, B, C, is equal to
the equiangular folid made from B.
Let E be a ſolid angle contained under the plain angles DEF,
GEF, GED; make ĞE, EF, DE, cach equal to B; and com-
pleat the ſolid parallelopipedon EK. Again, put- LM equal to
A; and at the point L, with the right line LM, make a ſolid
angle contained under the three plain angles NLX, XLM,
MLN, equal to the ſolid angle E a.
a 24
Then, becauſe LM, LN, EF, EG, or ED, and LX, are e-
qual to the three proportional lines A, B, C; therefcre, as LM
is to EF, fo is GE to LX; therefore the ſides about the equal
angles GEF, MLX are reciprocally proportional; and the pa-
rallelograms MX, GF, equalb; and ſince the two plain angles b 14.6.
GEF, XLM, are equal, and from the points N, D, are drawn
the equal perpendiculars NL, DE, ar right angles to the plain
palling through XLM, GEFC; therefore the folids LH, EK,
have the ſame altitude; but their baſes MX, GF, are equal;
therefore the ſolid HL is equal to the ſolid EK d; but HL is made
d 33.
of the three right lines A, B, C, and KE of the right line B.
Wherefore, &c.
ܪ
ܪ
at
C 35
d.
1
PRO P. XXXVII. THEOR.
IF
F four right lines are proportional, the ſimilar ſolid parallelopi-
pedons deſcribed from them fall be proportional ; and if the
ſimilar folid parallelopipedons be proportional, then the right lines
they are deſcribed from ſhall be proportional.
Let the four right lines AB, CD, EF, GH, be proportional,
viz. AB to CD, as EF is to GH; and let KA, LC, ME, NG,
be the ſimilar parallelopipedons deſcribed from them; then KA
;
is to LC as ME is to NG.
For, becauſe the ſolid parallelopipedon KA is ſimilar to LC,
KA is to LC in the triplicate proportion of AB to CDa. Fora 33.
the ſame reaſon, the folid ME is to NG in the triplicate pro-
portion of EF to GH; but AB is to CD as EF to GH ; there b 11. si
fore AK is to LC as ME to NG 6. And, if AK be to LC as
ME to NG, then AB is to CD as EF is to GH. For, becauſe
AK has to LC a triplicate proportion of AB to CD, and ME to
NG, a triplicate proportion of EF to GH ; therefore AB is to
CD as EF to GH5, Wherefore, &c.
PRO P
128
Τ Η Ε Ε L Ε Μ Ε Ν Τ S
Book XI.
PRO P. XXXVIII. THEO R.
T
"
IF
F a plain be perpendicular to a plain, and a line be drawn from
a point in one of the plains perpendicular to the other plain,
that perpendicular ſhall fall in the common ſection of the plain.
3 def. 4.
b def. 3.
Let the plain CD be perpendicular to AB; and, in the plain
CD, any point E be taken, from which let fall the perpendicu-
lar EG ; then EG ſhall be perpendicular to the common ſection
AD.
For, if not, let it fall without the common fection, as EF, and
draw FG in the plain EGF, perpendicular to AD“; then, be-
cauſe FG is perpendicular to the plain CD, and the right line
EG in the plain CD touches it, the angle FGE is a right angleb;
but EF is alſo at right angles to the plain AB; therefore the
angle EFG is a right angle: Therefore two angles in the triangle
FGE are equal to the two right angles ; which cannot be €;
therefore the right line drawn from the point E perpendicular
to AB, does not fall without the line AD therefore muſt fall
on it. Wherefore, &c.
.
CI7. I.
و
ވެ
PRO P. XXXIX. THEOR.
I ,
F the ſides of the oppoſite plains of a ſolid parallelopipedon
be divided into two equal parts, and plains be drawn thro'
their common Jeations, the common ſection of theſe plains, and the
diameter of the ſolid parallelopipedon Mall biſect each other.
>
22. I.
Let the ſides of CF, AH, the oppoſite plains of the ſolid paral.
lelopipedon AF, be each bitected in the points K, L, M, N, X,
O,P, R; let the plains KN, XR, be drawn through the ſections ;
and let YS be the common ſection of the plains, and DG the
diameter of the ſolid ; then YS, DG, will bifect each other.
For, join DY, YE, BS, SG ; then, becauſe DX is parallel
to OE, the alternate angles DXY, YOE, are equal“; and, be-
,
cauſe DX is equal to OE, and YX to YO, and they contain ea
qual angles, the bafe DY is equal to YE, and the triang
DXY to YOE, the angles in the one equal to the angles of t
other, each to each”, viz. the angle XYD equal to the ang.
OYE, and OEY to XDY ; therefore DYE is a right line
For the ſame reaſon, BSG is a right line, and BS equal to SG.
T
54. I.
b
14, I.
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OF EUCLIV.
129
y
e 2.
f .
Then, becauſe CA is equal and parallel to DB, and to EG4, DB Book XI.
is parallel to EG, and DE to GB d; but D, E, G, B, Y, S,
are points taken in each of them. Join DG, YS; then DG, d 34. 1.
ᎠG; ,
YS, are in one plain f; and, ſince DE is parallel to GB, the
angle EDT is equal to BGT «; but the angle DTY is equal to the a 29. s.
angle GTS 8; therefore DTY, GTS, are two triangles, having & 15. 1.
the angles YDT, DTY, equal to the two angles SGT, GTS,
each to each, and the fide YD equal to GS; therefore DT is en
qual to TG, and YT to TS. Wherefore, &c.
h 26. 1.
PRO P. XL.
T H E O R.
IF
F two triangular priſms of equal altitudes, the baſe of one of
which is a parallalelogram, and the other a triangle, and, if
the paraiielog: um be double the triangle, the prifms are equal to
each other.
Let ABCDEF, GHKLMN, be two priſms of equal altitude;
let the parallelogram AF be the baſe of the one, and the triangle
GHK the baſe of the other; and, if AF be double GHK, the
priſms are equal.
For, compleat the folids AX, GO; then, becauſe AF is
double GH, and the parallelogram HK double the triangle
GHK"; the parallelograms AF, HK, are equal; therefore the a 41. I.
folids AX, GO, are equal 0; but the half of equal things are b 31.
equal; therefore the priſm GHKLMN is equal to the priſm
ABCDEF; for each is half the folids GO, AXWherefore, C 28.
&c.
A
R
ge
Τ Η Ε
}
Τ Η Ε
E L E
Ε Μ Ε Ν Τ 8
M E T S
N
.
OF
1
E
U U C
L I D.
BOOK
XII.
1
P R O P. 1. THE O R.
Book XII
S
IMILAR polygons inſcribed in circles, are to one another as
the ſquares of the diameters of the circles.
;
b
Let ABCDE, FGHKL, be circles, in which are inſcribed the
ſimilar polygons ABCDE, FGHKL ; let BM, GN, be the
diameters of the circles ; then the polygon 'ABCDE is to
“
FGHKL as BM ſquare is to GN ſquare.
For, Join BE, AM, GL, FN ; then, becauſe the polygons.
are ſimilar, the angles BAE, GFL, are equal; and BA is to
AE as GF to FL; wherefore the triangles ABE, FGI., are e-
a 6.6.
quiangular“; that is, the angle AEB equal to FLG, and the
fides about them proportional; but the angle AMB is equal to
b 21. 3. A EBD, and FLG to FNG b: therefore the angle AMB is equal
to FNG, the angle BAM to GFN“, and the remaining angle
ABM to FGN: Therefore the triangles AMB, GFN, are equi.
d 4. 6. angular, and BM is to GN as BA is to GFd; but the triangle
ABM is to the triangle FGN in the duplicate ratio of AB to
e 19. 6.
GF, and the polygon ABCDE is to the polygon FGHKL in
the duplicate ratio of AB to GFf; but the triangle ABM is to
the triangle FGN in the duplicate ratio of BM to GN; there-
fore the polygon ABCDE is to the polygon FGHKL in the dų.
8 22. 5. 6
plicate ratio of BM to GN.' Wherefore, &c.
C 31. 3
с
3
I
e
f
20. 6.
8
}
P R O P
+
}
2
OF EUCLI D.
131
Book XII
PROP. I. BOOK X. L EMMA,
IF F there be two unequal magnitudes, froin the greater be taken
a part greater than its half, and from the remainder a part ta.
ken greater than its half ; this may be done till the magnitude re-
maining be leſs than any propoſed magnitude.
m
Let AB and C be two unequal magnitudes, of which AB is
the greater ; from AB let a part BH be taken greater than the
half, and from the remainder AH a part KH greater than its
half; and ſo on, till the remaining magnitude, which let be
AK, be leſs than the aſſigned magnitude C. Let C be multi-
plied till it become greater than AB, which let be DE, and di-
vide it into the parts DF, FG, GE, each equal to C. Then,
becauſe DE is greater than AB, and the part EG taken from it
leſs than the half thereof, and the part BH greater than the
half of AB, there remains DG greater than AH. Again, be-
caufe GD is greater than HA, and GF, half of GD, is taken
from it; and if from HA be taken HK greater than the half of
HA, there will remain FD greater than KA; but FD is equal
to C; therefore KA is leſs than C. Which was required.
PROP. II. THEO R.
CH
IRCLES are to each other as the ſquares of their diame.
ters.
Let ABCD, EFGH, be circles, whoſe diameters are BD,
FH; then, as the ſquare of BD is to the ſquare of FH, ſo is the
circle ABCD to the circle EFGH. If not, the circle ABCD
will be to ſome figure either leſs or greater than the circle
EFGH.
Firſt, let it be to a figure S, leſs than the circle EFGH, in
which inſcribe the fquare EFGH, which will be greater than
half the circle. For, if tangents are drawn to the circle, thro'
the points E, F, G, H, the ſquare EFGH will be half the ſquare
defcribed about the circle, but the circle is leſs than the ſquare
deſcribed about it; therefore the ſquare EFGH is greater than
half the circle. Let the circumferences EF, FG, GH, HE, be
biſected in the points K, L, M, N, and join EK, KF, FL, LG,
GM,
>
1
!
132
T HE E LE M ENTS
2 41. I.
2
;
b Lem.
C 1.
d hyp.
CII. 5.
f 14. 5.
Book XII GM, MH, HN, NE, and if tangents are drawn from the points
K, L, M, N, and parallelograms compleated upon EF, FG,
GH, and HE; then each of the triangies EKF, FLG, GMH,
HNE, will be equal to half the parallelogram, and therefore
greater than half the ſegment of the circle it ftands in ; if the rea
maining ſegments are biſected, and triangles drawn, as before
and this be continued till the ſegments are leſs than the exceſs by
which the circle EFGH exceeds the figure Sb. Let theſe be
.
the ſegments cut off by the right lines EK, KF, FL, LG, GM,
MH, HN, NE; then the remaining polygon EKFLGMHN
will be greater than the figure S.
Deſcribe the polygon AXBOCPDR, in the circle ABCD,
fimilar to the polygon EKFLGMHN'; then, as the ſquare of
BD is to the ſquare of FH, fo is the circle ABCD to the figure
Sd, and, as the polygon AXBOCPDR to the polygon
EKFLGMHN, ſo is the circle ABCD to the figure S'; but the
circle ABCD is greater than the polygon in it; therefore
the figure S is greater than the polygon EKFLGMHN f; but
it is leſs ; which is abfurd; therefore the ſquare of BD to the
ſquare of FH is not as the circle ABCD to fome figure leſs than
the circle EFGH. In like manner, it is proved that the ſquare
of FH to the ſquare of BD is not to the circle EFGH as ſome
figure leſs than the circle ABCD.
Laſtly, the ſquare of BD to the ſquare of FH, is not as the
circle ABCD to ſome figure greater than the circle EFGH.
For, if poſſible, let it be to the figure T, greater than the circle
EFGH; then, inverſely, the ſquare of FH is to the ſquare of
BD as the figure T is to the circle ABCD; but, becauſe T is
greater than the circle EFGH, T will be to the circle ABCD
as the circle EFGH is to ſome figure leſs than the circle ABCD,
which is proved impoffible; therefore the ſquare of BD to the
ſquare of FH is not as the circle ABCD to ſome figure leſs than
the circle EFGH, nor to one greater; therefore, as the ſquare of
BD is to the ſquare of FH, ſo is the circle ABCD to the circle
EFGH. Wherefore, &c.
PRO P. III.
THE O R.
En
VERY pyramid having a triangular baſe, may be divided
into two pyramids, equal and ſimilar to one another, having
triangular baſes, and ſimilar to the whole pyramid ; and into two
equal priſms; which two priſms are greater than the half of the
whole pyramid.
Let
1
OF EUCLI D.
133
!
n
9
b
34. I.
>
C 29. 1.
3
Book XI
Let there be a pyramid, whoſe baſe is the triangle ABC, and
vertex the point M, then the pyramid ABCM may be divided in-
to two pyramids, equal and ſimilar to one another, having tri-
angular bafes, and fimilar to the whole; and into two equal
priſms; which two priſms are greater than the half of the whole
pyramid.
For, biſect AB, BC, AC, MA, MB, MC, in the points
E, N, G, H, K, L; join EH, EG, EK, EN, HG, HK,
HL, KL, KN, NG ; then, becauſe AE is equal to EB, and
AH to HM, EH is parallel to MB , and KH to AB b; but Al å 5. 6.
is equal to KH; for each is equal to EB; therefore the two fides
AE, AH, are equal to the two fides KH, HM, the angle MHK
equal to HAE“; therefore the baſes EH, MK, are equal", an d *. 1.
the triangle AEH equal and ſimilar to KHM. For the ſame
reaſon, AHG is equal and ſimilar to MHL. And, becauſe AE,
AG, are equal and parallel to HK, HL, each to each, the angle
KHL is equal to EAG, and the baſe KL to EG"; therefore the c 10. 11.
triangles KML, EHG, are equal and fimilar; therefore the py-
ramid, whoſe baſe is the triangle AEG, and vertex the point H,
is equal and Gimilar to the pyramid whoſe baſe is the triangle
HKL, and vertex the point Mf; and, becauſe HK is parallel to def. 7o.
AB, the ſide of the triangle AMB, the triangles AMB, 11.
MHK, are ſimilars. For the ſame reaſon, the triangle MBC is 8 2. 6.
.
ſimilar to MKL, and the triangle AMC to MHL; but the angles
KHL, BAC, are equal, and the triangles ſimilar e; therefore
the pyramid ABCM is ſimilar to HKLM; but the pyramid
AEHG is proved ſimilar to HKLM, therefore ſimilar to one . 28.
28. C.
nother, and ſimilar to ABCM.
Again, becauſe BN is equal to NC, the parallelogram BG is
double the triangle GNC i ; therefore the priſm contained byi 41. [.
the two triangles BKN, EHG, and the three parallelograms
BG,BH, KG, is equal to the priſm contained by the two triangles
NGC, KHL, and the three parallelograms KG, GL, NL, the one
of which is conſtitute upon the parallelogram BG, and oppoſite to
it the right line KH; the other upon the triangle GNC, and op-
poſite to it the triangle KHL; and the parallelogram BG is
double the triangle GNC, and have the ſame altitude; therefore
they are equalk; but either of theſe priſms is greater than the k 40. II.
pyramid AEGH, or HKLM; for the priſm EBNGHK is great-
er than the pyramid EBNK, which is equal to the pyramid
AEGHT, or HKLM; wherefore the priſm EBNGHK is greater
than the pyramid AEGH, or HKLM; therefore the prifm
GNCHKL, is likewiſe greater than the pyramid HKLM;
but the prifms are equal; therefore, together, are greater than
the two pyramids ; therefore the whole pyramid is divided into
two equal pyramids fimilar to the whole, and to one another, and
into
134
THE ELEMENTS
Book XII into two equal priſms, which two priſms together are greater
than half the pyramid.
PROP. IV. THEOR.
If there are two pyramids, of the ſame altitude, having trian.
gular baſes, and each of them divided into two pyramids equal
to one another, and ſimilar to the whole, and into two equal priſms;
and, if each pyramid be divided in the ſame manner, and this be done
continually; then, as the baſe of the one pyramid is to the baſe of
the other, so are all the priſms in the one pyramid to all the priſms
in the other, being equal in number.
1
a 4. 6.
b 22. 6.
Let there be two pyramids of the ſame altitude, having the
triangular baſes ABC, DEF, and vertices the points M, H, and
each of them divided into two pyramids, equal to one anothers
and ſimilar to the whole, and into two equal priſms; and if, in
like manner, each of the pyramids made by the former diviſion
be ſuppoſed divided, and this be done continually; then, as the
baſe ABC is to the baſe DEF, ſo are all the priſms in the pyra-
mid ABCM to all the priſms in the pyramid DEFH, being equal
in number.
For, let the pyramid DEFH be conſtructed ſimilar to the py-
ramid ABCM; then, all the triangles deſcribed in the baſe
ABC being ſimilar to the whole, and to one another; and alſo
thoſe in DEF, being equal in number to the triangles in ABC;
then ABC will be ſimilar to NGC, and DEF to RQF; and, as
BC is to NC, fo is EF to FQ; therefore ABC is to NGC as
DEF is to ROFb. And, altern. as ABC is to DEF, ſo is NGC
to RQF; but, as NGC is to RQF, ſo is the priſm GNCLHK
to the priſm RQFYST°: But the two priſms in the pyramid
ABCM are equal to one another d, as alſo the two priſms in
the pyramid DEFH; wherefore the priſm, whoſe baſe is the pa-
rallelogram EGNB, and oppoſite baſe the right line KH, is to
the priſm, whoſe baſe is the triangle NGC, and oppofite baſe
the triangle HKL, ſo is the priſm whoſe baſe is the paralle-
gram EPRQ, and oppoſite bale the right line ST to the
priſm whoſe baſe is the triangle RQF, and oppoſite to it
the triangle STY , compound. as the priſms EBNGKH,
GNCLHK, together, are to the priſm GNCLHK, ſo the priſms
PEQRST, ROFSTY, together, are to RQFSTY; altern. as
the priſms EBNGKH, GNCLHK, together, are to the priſms
PEORST, RQFSTY, together, ſo the priſm GNCLHK to
ROFSTY ; but the priſm GNCLHK is to the priſm ROFSTY
C 32. and
с
28. 11.
& 3.
C
j
as
OF EU CLI D.
135
as the baſe GNC to the baſe RQF and fo is the baſe ABC to Book XII
the baſe DEF; therefore, as the bafe ABC is to the baſe DEF,
lo are the two priſms in the pyramid ABCM to the two priſms in
the pyramid DEFH. For the ſame reaſon, the priſms in the py.
ramids HKLM, STYH, or any other pyramids made by any of
the former diviſions, are to each other as th:ir baſes; wherefore,
all the priſms in the pyramid ABCM are to all the priſms in the
pyramid DEFH as the baſe ABC to the DEF. Wherefore, &c.
PRO P. V. and VI. THE O R.
P
YRAMIDS of the ſame altitude, having triangular or
polygonous baſes, are to one another as their bafes.
24:
Firſt, let ABCM, DEFH, be pyramids of the ſame altitude,
having the triangular baſes ABC, DEF; then the pyramid
ABCM is to the pyramid DEFH as the baſe ABC is to DEF;
and ſo are any number of pyramids to their triangular baſes.
If not, let the baſe ABC be to the baſe DEF as the pyramid
ABCM is to ſome folid 2, leſs than the pyramid DEFH, which
divide into two pyramids equal to each other, and into two
friſms which are greater than half of the whole pyramid ; and,
if the pyramids made by the former diviſion be divided in the
ſame manner, till ſome pyramids in the pyramid DEFH is found
leſs than the exceſs by which the pyramid DEFH exceeds Z.
Let thefe pyramids bé DPRS, STYH. Let the pyramid
ABCM be divided into the fame number of ſimilar parts, as the
pyramid DEFH; then are the priſms in the pyramid ABCM to
the priſms in the pyramid DEFH ?, as the baſe ABC is to
a
DEF; but the baſe ABC is to the baſe DEF as the pyramid
ABCM to the ſolid Z b; therefore the pyramid ABCM is to the b hyp.
folid Z as the priſms in ABCM to the priſms in DEFH; but
ABCM is greater than the priiins in it; therefore the ſolid Z
is greater than the priſms in DEFH°; and likewiſe leſs b; which c 14. s.
is abſurd ; therefore the baſe ABC is not to the bale DEF as the
pyramid ABCM to ſome ſolid leſs than DEFH. For the ſame
reaſon, the baſe DEF is not to the baſe ABC as the pyramid
DLFH to ſome folid leſs than ABCM ; but ABC is not to
DEF as ABCM is to ſome folid I, greater than DEFH. For,
if poſſibie, DEF is to ABC as I to ABCM d; but the folid lis d inverſ.
greater than DEFH; then, as I is to ABCM, ſo is DEFH to
ſome folid leſs than ABCM ; which is proved abſurd; therefore
ABC to DEF is not as ABCM to Tome ſolid greater than
DEFH; but it was alſo proved not to be to ſome ſolid leſs than
DEFH; therefore ABC is to DEF as ABCM is to DEFH.
For
ز
ز
ز
ز.
136
Τ Η Ε Ε L Ε Μ Ε Ν Τ S
Book XII
For the ſame reaſon, in the pyramids ABCDEM, FGHKLN,
fig. 1. the pyramid ABCM is to the pyramid ACDM as the
bite ABC is to the baſe ACD ; and ACDM is to ADEM as
ACD is to AED, therefore the whole ABCDEM is to ABCM
as ABCDE to ABC d. For the ſame reaſon, as FGHKLN is
to FGHKL as FGHN is to FGH; if ABCDE is equal to
FGHKL, the pyramid ABCDEM is equal to FGHKLN; if
greater, greater, and, if leſs, leſs; therefore, as the baſe
ABCDE is to the baſe FGHKL, fo is the pyramid ABCDEM
to the pyramid FGHKLNC; Wherefore, &c.
d 12. 5.
ز
e def. s.
PRO P. VIL. THE O R.
VERY priſm, having a triangular baſe, may be divided
into three pyramids, equal to one another, and having trian-
gular baſes.
F
a 34. I.
Let there be a priſm, whoſe baſe is the triangle ABC, and
the oppoſite baſe to that the triangle DEF; then the priſni
ABCDEF may be divided into three equal pyramids, having
triangular baſes.
For, join BD, EC, CD; then, becauſe ABCD is a paralle
logram, whoſe diameter is BD, the pyramid whofe baſe is the
b s. and o.
triangle ABD, and vertex the point C, is equal to the pyramid
whofe baſe is the triangle EBD, and vertex the point CD; but
the pyramid whoſe baſe is the triangle EBD, and vertex the
point C, is equal to the pyramid whoſe baſe is the triangle
EBC, and vertex the point D ; for they are contained by the
fame plains; therefore the pyramid whoſe baſe is the triangle
ABD, and vertex the point C, is equal to the pyramid whoſe
baſe is the triangle EBC, and vertex the point D.
Again, becauſe FCBE is a parallelogram whoſe diameter is
CE, the triangle ECF is equal to the triangle CBE “; therefore
the pyramid whoſe baſe is the triangle BEC, and vertex the
point D, is equal to the pyramid whoſe baſe is the triangle CEF,
and vertex the point Db: But the pyramid whoſe baſe is the tri-
angle BEC, and vertex the point D, has been proved equal to
the pyramid whoſe baſe is the triangle ABD, and vertex the
point C; therefore, alſo the pyramid whoſe baſe is the triangle
CEF, and vertex the point D, is equal to the pyramid whoſe
baſe is the triangle ABD, and vertex the point C; therefore
the priſm ABCDEF is divided into three pyramids, equal to one
another, and having triangular baſes. And, becauſe the pyra.
mid whoſe baſe is the triangle ABD, and vertex the point ,
is the ſame with the pyramid whoſe baſe is the triangle ABC,
and
1
OF EUCLI D.
137
1
and vertex the point D, for they are contained by the fame Book XII
plains; and the pyramid whoſe baſe is the triangle ABD, and
vertex the point C, has been proved to be a third part of the
priſm, having the fame baſe, viz. the triangle ABC, and the
oppoſite baſe the triangle DEF: Which was required.
COR. I. Hence every pyramid is a third part of a priſm, ba.
ving the ſame baſe, and an equal altitude; for, if the bale of a
prifm be of any other figure, it can be divided into priſms,
having triangular baſes.
II. Priſms of the ſame altitude are to one another as their
baſes.
a
1
1
P = 0 P. VIII.
THE OR.
S triplicate ratio of
IMILAR pyramids, having triangular baſes, are in the
triplicate ratio of their homologous filles.
>
Let the two pyramids whoſe bafes are ABC, DEF, and ver-
tices the points G, H, be ſimilar and alike ſituate, then the py- •
ramids ABCG, DEFH, are to one another in the triplicate ra-
tio of BC to EF.
For, compleat the parallelopipedons BGML, EHPO, then
each contain two equal priſms, having triangular baſes"; and a # 23. II.
pyramid is one third of a priſm, having the ſame baſe and alti-
tude b; but ſimilar folid parallalelopipedons are to one another b 7.
in the triplicate ratio of their homologous fides, and parts have c 33. 11.
the ſame proportion as their like multiples d; therefore the pyra-d 15. s.
mids ABCG, DEFH, are to one another in the triplicate ratio
of their homologous fides. Wherefore, &c.
Cor. Hence ſimilar pyramids, having polygonous baſes, are
to one another in the triplicate ratio of their homologous
ides.
C
n
PRO P. IX. THE O R.
HE baſes and altitudes of equal pyramids, having triangu-
lar baſes, are reciprocally proportional ; and thoſe pyrı-
mids, having triangular baſes, whoc bafes and altitudes are reci.
procally proportional, arc equal.
THE
S
Let
138
Τ Η Ε Ε L Ε Μ Ε Ν Τ S
.
and 7.
b
Book XII Let the pyramids whoſe triangular baſes are ABC, DEF, and
vertices the points G, H, be equal; then the baſe ABG is to
the baſe DEF, as the altitude of the pyramid DEFH is to the
altitude of the pyramid ABCG.
For, compleat the folids BGML, EHPO, then, becauſe the
pyramids ABCG, DEFH, are equal, the folids ABGML,
a 28. 11. EHPO, are equal; but equal folid parallelopipedons have their
bafes and altitudes reciprocally proportional b; therefore the py-
b 34. 11.
ramids ABCG, DEFH, have their baſes and altitudes recipro-
cally proportional '; and, if their bafes and altitudes are reci.
procally proportional, they are equal. For, the ſame conſtruction
remaining, the folid parallelopipedons, whoſe baſes and altitudes
are reciprocally proportional, are equal; therefore pyramids of
the ſame altitudes with the ſolids, having their baſes and alti-
tudes reciprocally proportional, are likewiſe equal. Where-
fore, &c.
C 15. S
ý
PRO P. X. THE O R.
OR
1
Е.
VERY cone is the third part of a cylinder, having the ſame
baſe, and an equal altitude.
9
a 2.
Let there be a cone and cylinder, having the ſame baſe, viz.
the circle ABCD, and their altitudes equal, then the cone is
one third of the cylinder; that is, the cylinder is triple the cone.
If not, it will be either greater or leſs than triple the cone. Firſt,
let it be greater, and let a polygon AEBFCGDH be inſcribed
in the circle ABCD, and let the ſmall ſegments AE, EB, BF,
FC, CG, GD, DH, HA, the exceſs by which the circle ex-
ceeds the polygon, be leſs than any aſſigned magnitude; and, up-
on the circle and polygon let a cylinder and pyramid be deſcri-
bed, of the ſame altitude with the cone ; and, upon the remain-
ing ſegments, the remaining parts of the cylinder, which let be
leſs than the exceſs by which the cylinder exceeds triple the cone;
therefore the priſm whoſe baſe is the polygon AEBFCGDH, and
altitude the ſame of the cone, is greater than triple the cone;
but the priſm is triple the pyramid of the ſame baſe and altitude
of the cone b; therefore the pyramid is greater than the cone,
and likewiſe leſs, as included in it; which is abſurd ; therefore
the cylinder is not greater than triple the cone, neither is it leſs
for then, inverſely, the cone would be greater than one third of
the cylinder ; for, the ſame conſtruction remaining, the pyramid,
whoſe baſe is the polygon AEBFCGDH, and vertex the ſame
of the cone, is greater than one third of the cylinder ; but the
ز
b 7.
b
ܪ
pyramid
:
OF EUCLI D.
139
pyramid is one third of the priſm conſtitute on the ſame baſe, Book XII
and having the ſame altitude; therefore the pyramid whoſe baſe
is the polygon AEBFCGDH, and altitude the ſame of the cone,
is greater than the cone whoſe baſe is the circle ABCD ; and
likewiſe leſs, as contained in it; which cannot be; therefore the
cylinder is not leſs than triple the cone. Therefore, ſince nei-
ther greater nor leſs, it muſt be triple the cone. Wherefore,
&c.
و
PROP. XI. THEO R.
Cohen
ONES and cylinders, of the ſame altitude, are to one ano-
ther, as their baſes.
Let there be cones and cylinders of the fame altitude, whoſe
are the circles ABCD, EFGH, and axes KL, MN, and
diameters of their baſes AC, EG; then, as the circle ABCD is
to the circle EFGH, ſo is the cone AL to the cone EN, If not,
the circle ABCD is to the circle EFGH, as the cone AL is to
ſome ſolid greater or leſs than the cone EN. Firſt, let it be to a
folid X leſs than the cone; and let the ſolid I be equal to the exceſs
of the cone EN above the ſolid X; then the cone EN is equal to the
ſolid I and X together. Let a polygon HOEPFRGS be inſcri-
bed in the circle EFGH, of which the remaining circumferences
HO, OE, EP, PF, FR, RG, GS, SH, are leſs than any
aſſigned magnitudes. Upon the polygon HOEPFRGS ler a
pyramid be deſcribed, of the fame altitude with the cone, and
let the remaining fegments of the cone deſcribed upon the cir-
cumferences HO, OE, EP, PF, FR, RG, GS, SH, and vertex the
fame as the pyramid be leſs than the folid!; therefore the pyra-
mid HOEPFRGS, and altitude the fame of the cone, will be
greater than the folid X.
Upon the circle ABCD let the polygon DTAYBQCV be
deſcribed fimilar and alike fituate to HOFPFRGS, and let a
pyramid EN be erected, of the ſame altitude as the cone AL;
but the polygons DTAYBQCV, HOEPFRGS, are to one ano-
ther as the ſquares of their diameters AC, EG“, and the circlesa I.
a
ABCD, EFGH, are to one another as the ſquares of their dia-
meters AC, EGb; therefore, as the circle. ABCD is to theb 2.
circle EFHG, ſo is the polygon DTAYBQCV to the polygon
HOEPFRGS ; but, as the circle ABCD is to the circle EFGH,
fo is the cone AL to the folid X ; therefore the poiygon
DTAYBQCV is to the polygon HOEPFRCSC as the concc 13. $,
AL is to the folid Xd; but the pyramid DTAYBOCVL is tod hyp.
the
>
140
THE ELEMENTS
ܪ
>
Book XII the pyramid HOEPFRGSN as their bafes ®; therefore the pyram
mid DTAYBQCVL is to the pyramid HOEPFRGSN as the
e s. and 6. cone AL is to the folid X; but the pyramid is greater than the
ſolid X, and the cone AL graeater than the pyramid in it;
therefore, likewiſe the cone EN is greater than the pyramid in
it; but the pyramid in the cone EN is greater than X; therefore
the cone EN is much greater than X; but it was put lefs ; which
is abfurd ; therefore the circle ABCD, to the circle EFGH, is
not as the cone AL to a ſolid leſs than the cone EN: and it is
proved, in the ſame manner, that the circle EFGH is not to the
circle ABCD, as, the cone EN is to a ſolid leſs than the cone
AL.
Again, the circle ABCD to the circle EFGH, is not as the cone
AL to a ſolid Z greater than the cone EN; then, inverſely, as
the circle EFGH is to the circle ABCD, ſo is the ſolid Z to the
cone AL; but the ſolid Z is greater than the cone EN. Then,
as the ſolid Z is to the cone AL, ſo is the cone EN to ſome fo-
lid leſs than the cone AL; therefore, as the circle EFGH is
to the circle ABCD, ſo is the cone EN to ſome folid leſs than
the cone AL; which is impoffible; therefore the circle ABCD
to the circle EFGH is not as the cone AL to ſome ſolid greater
or leſs than EN, therefore, to the cone EN; but, as cone is to
cone, ſo is cylinder to cylinderf, Wherefore, &c.
th
W
f Is. 5.
:
PRO P. XII.
THE O R.
SIMILAR cones and cylinders are to one another, in the tri-
plicate ratio of the diameters of their baſes.
Let there be ſimilar cones and cylinders, whoſe baſes are the
circles ABCD, EFGH, their diameters BD, FH, and axes of
the cones and cylinders KL, MN; then the cone whoſe baſe is
the circle ABCD, and vertex the point L, to the cone whoſe
baſe is the circle EFGH, and vertex the point N, hath a tripli-
cate ratio of BD to FH.
For, if the cone ABCDL be not to the cone EFGHN, in
the èriplicate ratio of BD to FH, let it be in the triplicate ratio to
ſome ſolid greater or leſs than the cone EFGHN. Firſt, let it
be to a ſolid X, leſs than the cone EFGHN, and let the polygon
EOFPGRHS be the greateſt polygon poſſible inſcribed in the
circle EFGH; that is, that the exceſs of the circle above the in-
fcribed polygon be leſs than any aſſigned magnitude; upon the
Folygon LOFPGRHS let a pyramid be deſcribed, of the ſame
altitude
OF EUCLI D.
141
2
and is. 5.
'c 8.
altitude of the cone, and the fegments of the cone deſcribed upon Book XIL
the ſegment of the circle, greater than the polygon, be leſs than
the exceſs by which the cone EFGHN exceeds the folid X; then
the pyramid deſcribed on the polygon EOFPGRHS, of the ſame
altitude as the cone, is greater than the ſolid X. Let the poly-
gons ATBYCVDQ be inſcribed in the circle ABCD, ſimilar
to the polygon EOFPGRHS, and upon it deſcribe a pyramid a 18. S.
a
6,
of the ſame altitude of the cone. For, upon the polygon
EOFPGRHS, fuppofe prifms erected, of the fame altitude of
the cone; then there poſms are to one another as their baſes b. b 32. 11.
For the ſame reaſon, the priſms deſcribed on the polygon
ATBYCVDQ; equiangular to thoſe on the polygon EOPGRHS,
and of the fame altitude of the cone, are to one another as their
baſe ; but the baſes are ſimilar to one another; therefore the
equiangular priſms are likewiſe limilar, and likewiſe the
pyra.
mids; therefore the pyramids are to one another, in the tripli-
cate ratio of their homologous fides"; that is, of BD to FH;
but the cone ABCDL is to the ſolid X, in the triplicate ratio of
BD to FH ; therefore, as the cone ABCDL is to the ſolid X,
ſo is the pyramid ATBYCVDQL to the pyramid EOFPGRHSN;
but the cone is greater than the pyramid EOFPGRHSN d; buid 14. So
it is proved leſs; which is abſurd; therefore the cone ABCDL
has not to a ſolid leſs than the cone EFGHN, a triplicate ratio
of BD to FH. For the ſame reaſon, the cone EFGHN has not
to ſome ſolid leſs than the cone ABCDL a triplicate ratio of
FH to BD.
Again, the cone ABCDL has not to a ſolid Z, greater than
EFGHN, a triplicate ratio of BD to FH ; for, then, inverſely,
the folid Z has to the cone ABCDL a triplicate ratio of FH to
BD; but the ſolid Z is greater than EFGHN; therefore the folid
Z, to the cone ABCDL, is as the cone EFGHN to ſome ſolid
leſs than the cone ABCDL ; therefore the cone EFGHN, to
ſome lolid leſs than the cone ABCDL, has a triplicate ratio of
FH to BD; but it is proved that it has not; therefore the
cone ABCDL, to
ſolid
greater
or leſs than the cone
EFGHN, has not a triplicate ratio of BD to FH ; therefore the
cones ABCDL, EFGHN, bave to one another the triplicate ra-
tio of their baies BD to FH; and, as cone is to cone, ſo is cy-
linder to cylinder Wherefore, &c.
9
2
e 15. s:
PRO P.
4
1
I
142
THE E L E M E N T S
Book XII
PRO P. XIII.
THE OR.
IE a cylinder be divided by a plain parallel to the oppoſite plains,
then, as one cylinder is to the other cylinder, fo is the axis of the
one to the axis of the other.
>
ܪ
;
Let the cylinder AD be divided by the plain GH, parallel to
the oppoſite plains AB, CD, and meeting the axis EF in the
point K; then, as the cylinder BG is to the cylinder GD, fo is
the axis EK to KF.
For, let the axis EF be produced both ways to L and M; let
EL be taken any multiple of EK; and FM any multiple of FK;
through the points L, N, X, M, draw plains parallel to AB,
CD, and with the centers L, N, X, M, draw the circles
OP, RS, TY, VQ, each equal to AB; and compleat the cy-
linders PR, RB, DT, TO; then, becauſe the axis LN,
NE, EK, are equal, the cylinders PR, RB, BG, are equal a.
For the ſame reaſon, the cylinders HC, DT, TQ, are equal ;
therefore the cylinder PG is the ſame multiple of the cylinder
BG, that the axis LK is of EK. For the ſame reaſon, the
cylinder GQ is the ſame multiple of GD that KM is of
KF; therefore, if KL is equal to KM, PG will be equal to
GQ; if greater, greater, and, if leſs, leſs. Therefore, AH is
b def. 5. 5. to GD as EK is to HF b. Wherefore, &c.
a 11.
a
2
PRO P. XIV.
THE O R.
C
O'NES and cylinders, conſtituted upon equal baſes, are to
one another as their altitudes.
a 11
Let the cylinders EB, FD, ſtand upon equal baſes AB, CD,
then the cylinder EB is to the cylinder FD, as the altitude GH
is to the altitude KL.
For, produce the axis KL to the point N, and put LN equal
to GH, and let the cylinder CM be drawn about the axis LN;
then the cylinders EB, CM, are to each other as their baſes a;
but their baſes are equal; therefore the cylinders EB, CM, are
equal; but the cylinders CM, FD, are as their axes LN, KL";
but the cylinders CM, EB, are equal; and their axes GH, LN,
likewiſe equal; therefore the cylinder EB is to the cylinder I'D
as the axis GH to the axis KL ; but, as the cylinder EB is to
$15.5, and the cylinder FD, ſo is the cone ABG to the cone CDK“; there-
fore, as the axis GH is to KL, ſo is the cone ABG to CDK
and ſo the cylinder EB to FD. Wherefore, &c.
PROP
b 13.
; ܝ
C
1.
OF EUCLI D.
143
ܕܐ
Book XII
PRO P. XV. THE O R.
T!
HE baſes and altitudes of equal cones and cylinders are reci-
procally proportional, and cones and cylinders whoſe baſe and
altitudes are reciprocally proportional, are equal to one ano-
ther
A
ز
a
>
ܪ
b 7. S.
Let the circles ABCD, EFGH, be the baſes of the equal
cones and cylinders, AC, EG, their diameters, and KL, MN,
their axes; compleat the cylinders AX, EO; then, as ABCD
is to EFGH, ſo is the altitude MN to KL.
For the altitudes KL, MN, are either equal or not. If equal,
the cylinders AX, EO, are likewiſe equal; then the baſes
ABCD, EFGH, are equal“; therefore the baſes ABCD, EFGH, a iri
are to one another as their altitudes; But, if the altitudes KL,
MN, are not equal, let one of them, as MN, be the greater, and
cut off PM equal to LK, and let the plain TYS, parallel to the
oppoſite plains ; cut the cylinder EO in the point P, and com-
pleat the cylinder ES ; then the cylinder AX is to the cylinder
ES as the cylinder EO is to the cylinder ES b; but the cylinder
AX is to the cylinder ES, as the baſe ABCD to the baſe
EFGH ~; and, as the cylinder EO is to the cylinder ES, fo is
the altitude MN to the altitude MP®; therefore the baſe ABCL C 13.
is to the baſe EFGH as the altitude MN is to the altitude KL;
therefore the baſes and altitudes of the cylinders AX, EO, are
reciprocally proportional.
And, if the baſes and altitudes of the cylinders AX, EO, are
reciprocally proportional, then the cylinders are equal; for, the
ſame conſtruction remaining, the baſe ABCD is to the baſe
EFGH, as the altitude MN is to the altitude KL; and the al-
titudes KL, MP, are equal; therefore the baſe ABCD is to the
baſe EFGH as the cylinder AX is to the cylinder ES“, and the
altitude MN is to the altitude MP as the cylinder EO is to
ES"; therefore the cylinder AX is to ES as EO is to ES; there-
fore the cylinder AX is equal to LO, and, becauſe cones are e g. s.
one third of the cylinder of the ſame baſe f and altitude, and 10.
parts have the ſame proportions as their like multiples 8; therea
fore the baſe and altitudes of equal cones and cylinders are reci-
procally proportional. And, &c.
f
5 15. 5.
PROP.
144
THE ELEMENTS
5
Book XII
PRO P. XVI.
PRO B.
1
TWO circles about the ſame center, to inſcribe
in the greater a
polygon of equal fides, even in number, that ſhall not touch the
leſer circle.
:
a 16. 3.
blem.
Let ABCD, EFGH, be two given circles, about the ſame
center K; it is required to inſcribe a polygon in the circle
ABCD, of equal fides, even in number, that ſhall not touch
the lefſer circle EFGH.
Through the center K draw the right line BD, and through
the point G draw AG at right angles to BD : produce AG to
C; then AC is a tangent to the circle EFGH, in the point G *;
bifect the circumference BAD, and, again, the half thereof, and
doing this, till a circumference is found leſs than AD, which
let be LD, draw LM perpendicular to BD, and produce it to
N; join LD, ND, Now, becauſe LN is parallel to AC, the
tangent of the circle EFGH, LN will not touch the circle
EFGH, and, much leſs, the lines LD, ND; and, if right lines
be applied in the circle, each equal to LD, there will be a po-
lygon inſcribed in the circle ABCD, of equal fides, and even
in number, that will not touch the leffer circle EFGH ; which
was to be done.
.
€ 29. 3.
7
PRO P. XVII. PRO B.
T! o deſcribe a ſolid polyhedron in the greater of two ſpheres, ha-
ving the ſame center, which ſhall not touch the ſuperficies of
the lefſer
ſphere.
a
11.
bis. 32
Let two ſpheres be ſuppoſed about the center A, it is required
to deſcribe a ſolid polyhe iron in the greater ſphere, not touching
the ſuperficies of the lefler ſphere.
Let the tphere be cut by lome plain paſſing through the cen-
a def. 14. ter, then the ſections will be circles a, and the circle deſcribed
a
by the half ſection will be a great circle b; which let be BEDC;
and FGH that of the leſſer ; and BD, CE, two of their diame-
ters, drawn at right angles to each other; let BD meet the leffer
circle in the point G; and draw GL a tangent to the leffer
circle in the point G; and join AL: In the greater circle BEDC
;
inſcribe a polygon that will not touch the lefſer circle FGH C;
let the ſides of the polygon, in the quadrant BE, be the right
Cio
lines
!
OF EUCLI D.
145
a
a
i
e
و
lines BK, KL, LM, ME, ſuch that each will ſubtend a lefs arch Book XII
than a line equal to the tangent Gt, then the right lines BK, KL, pred
LM, ME, will each be leſs than the tangent GL; and produce the
lines joining the points K, A,' N; and from the point A raiſe
AX perpendicular to the plain of the circle B.2DC, neering the
ſuperficies of the ſphere in x diet plains be drawn through d 12. Ir.
II
AX, BD, and AX, KN, which wil makro circies in the
fuperficies of the ſphere, and let 8X, CXN, be femicircies on
the diameters BD, KN; then, becauſc XA is perpendicular to
the plain of the circle BEDC, the femicircles BX, KIN, are
perpendicular to the ſame plain"; but the ſemicircles 399, e 18. II.
,
BXD, KXN, are equal, for they ſtand upon equal diameters
BD, KN, their quadrants BE, BX, KX, ali likwiſe be e.
qual ; therefore, as many ſides of the polygon as are in the qua-
drant BE, ſo many equal files y be v the quadrants BX,
KX; let theſe fides be BO, OP, PR, RX, KS, ST, TY, ;
and join SO, TP, YR; from the points :), 6, draw the pers
pendiculars OV, SQ, which will fall on BD, KN, the com-
mon fection f of the plain ; join VQ, then, tince the equil cir- f 33. 11.
cumferences BO, SK, are taken in the caug lengvircles BXD,
KXN; and, becauſe OV, SQ, are drawn perpendiculars from
them, they are equal; as alſo, BV, KQ; but BA, KA, are
equal; therefore AV is to VB as AQ is to OK; therefore VQ is
parallel to BK %, and OV is parallel to SQ; but it is proved 2-5 2. 6.
qual; therefore ZV, SO, are equal and paralleli; therefore OS,
i 33. I.
BK, are parallel k; but, becauſe BK is parallel to VQ, and AB k 9. 1.
equal to AK, AB is to BK as AV is to VO 5; and altern. AB is
to AV as BK is to Vk; but AB is greater than AV; therefore
BK is greater than VK; but VK is equal to OS; therefore BK
is greater than OS; join BO, KS, then OBKS is a quadrilateral
figure in one plain'. For the ſame reaſon, each of the quadrila-
teral figures SOPT, TPRY, and triangle YRS, are cach in one
plain ; therefore, if from the points U, S, P, T, K, Y, th the
point A, right lines are ſuppoled drawn), will conſtitute a poly-
hedrous figure within the circumferences BX, KX, confiting of
pyramids, whoſe baſes are KBOS, SOPT, TPRY, TRX, and
vertex the point A ; and if, in the ſame manner, pyramids be
conſtructed on the ſides KL, LM, ME, and on the other three
quadrants and cppofite hemiſphere, there will be conſtructed a
polyhedrous figure deſcribed in the fphere, compoſed of pyra-
mids whoſe bafes are equal and ſimilar to the forefuid quadrila-
teral figures, and triangle YRX, and vertex the point A.
But the polyhedron does not touch the fuperficies of the
ſpherc in which the circle IGH is. For, becauſe the quadrilateral
figure KESO is in one plain, and from the point A be drawn a
right line AZ perpendicular to tlie plain m, it will be at righe
T
angles
h 6
SI.
و
1
7: II.
+
>
>
.
.
1
17. II. Id
146
THE ELEMENTS
}
n
n def.
>
II,
077. I.
;
Book XII angles to all the right lines drawn in that plain "; join BZ,
ZK, then AZ will be perpendicular to BZ, ZK. But the
3. ſquares of AK, AB, are equal, and the ſquares of AZ, ZB, are
equal to the ſquare of AB, and the ſquares of AZ, ZK, are
equal to the ſquare of AK"; therefore the ſquares of. AZ, ZB,
are equal to the ſquares of AZ, ZK. Take the common ſquare
of AŹ from both, then the ſquares of EZ, ZK, are equal ; that
is, BZ equal to ZK. In like manner, lines drawn from Z to the
points o, S, inay be proved equal to BZ, ZK; therefore a circle
deſcribed about the center Z, with either of theſe diſtances, will
paſs through the points O, S, K, B; and, becauſe KBSO is a
quadrilateral figure inſcribed in a circle, and OB, KK, KS, are
equal, and OS leſs than BK, the angle BZK will be obtule
therefore BK is greater than BZ; but ĞL is greater than KB, and
therefore much greater than BZ ; and the ſquares of AG, GL,
are equal to the ſquare of AL, AB, or AK ; therefore the
fquares of BZ, ZA, are equal to the ſquare of AL; and the
ſquares of AZ, ZB, are equal to the ſquares of AG, GL; but
the ſquare of GL was proved greater than the ſquare of BZ;
therefore the ſquare of AZ is greater than the ſquare of AG;
that is, the right line AZ greater than AG; but AZ is perpen-
dicular to one of the baſes of the polyhedron; and AG reaches
the ſuperficies of the lefſer ſphere; therefore the polyhedron
does not touch the ſuperficies of the leſſer ſphere. Wherefore,
&c.
Cor. If a ſolid polyhedron is inſcribed in another ſphere, fi-
milar to that in BCDÉ, they ſhall be to one another in the tripli-
cate ratio of the ſquares of their diameters; for, the folids being
divided into pyramids, equal in number, and of the ſame order,
they will be fimilar ; and therefore to one another in the tripli-
cate ratio of their homologous fides P; that is, as AB drawn
from the center of the ſphere BCDE, to the ſemidiameter of the
other ſphere; but the ſemidiameters of ſpheres are as their dia-
meters 9; and one of the antecedents is to one of the conſequents
as all the antecedents to all the conſequents "; therefore, the
polyhedron in the one ſphere is to the fimilar polyhedron in the
other ſphere, in the triplicate ratio of their diameters.
a
و
Б 12.
PIs. 5.
r 12. 5
1
A
PRO P. XVIII. THEO R.
SPHERES are to one another in the triplicate ratio of their
Let
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Prop. 2
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ERRATA for Book XI. and XII.
P. XI. pr. 2. for def. 4.11. r. def. 4.1. pr. 21. for note àz r. a 20. pr. 24
for note, b 23.1 r. b 33. ' p. 123. l: 13. for P. r. PI,
B. XII. pr. 7. 1. S.for ABCD, r. ABED. p. 145. I. 34. for note 6.12. r. 7.1 It
-1, 25. for ZV r. QV. and 1.31. for note 7.15 1. 6.11.
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OF EUCLID.
147
t
1
>
4
2
.
b
с
Let ABC, DEF, be two ſpheres, and BC, EF, their diameters, Book XII
the ſphere ABC is to the ſphere DEF, in the triplicate ratio of
BC to EF. If not, let the ſphere ABC be to a ſphere GHK leſs
than the fphere DEF, in the triplicate ratio of BC to EF. Let
this ſphere GHK be inſcribed within the ſphere DEF; likewiſe,
in DEF, inſcribe a polyhedron, which ſhall not touch the ſuper-
ficies of the leffer ſphere GHK? In the ſphere ABC inſcribe a a 17.
polyhedron, ſimilar and alike fituate to that in DEF; then theſe
ſimilar polyhedrons are to one another in the triplicate ratio of
their dianieters BC, EF b; but the ſphere ABC, to the ſphere b cor. 17
GHK, hath a triplicate ratio of BC to EFC; therefore the ſphere c hyp.
ABC is to the ſphere GHK, as the polyhedron ABC to the fi-
milar polyhedron in DEF; but the ſphere AB is greater than
the polyhedron in it; therefore the ſphere GHK is likewiſe
greater than the polyhedron in DEF; but it is leſs, as contained
in it; which is abſurd; therefore the ſphere ABC, to the ſphere
leſs than DEF, has not a triplicate ratio of BC to EF. For the
ſame reaſon, the ſphere DEF, to a ſphere leſs than ABC, has not
a triplicate ratio of EF to BC. Again, the ſphere ABC, to a
{phere of LMN, greater than DEF, has not a triplicate ratio of
BC to EF. If it can, then, by inverf. the ſphere LMN, to the
,
ſphere ABC, ſhall have a triplicate ratio of the diameters EF to
the diameter BC; but the ſphere LMN is to the ſphere ABC as
the ſphere DEF to fome ſphere leſs than ABC, becauſe the
ſphere LMN is greater than DEF; therefore the ſphere DEF,
to a ſphere leſs than ABC, has a triplicate ratio of what EF has
to BC ; which is proved abſurd; therefore the ſphere ABC, to a
ſphere greater or leſs than DEF, has not a triplicate ratio of
what BC has to EF. Therefore ABC has to the phere DEF a
triplicate ratio of what BC has to EF : Which was to be demon-
ſtrated.
ܪ
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Τ Η Ε
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Τ Η Ε
+
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E L M
Ε L Ε Μ Ε Ν Τ S
L M N T S
0 F
P L AIN AND SPHERICAL
TRIGONOMETRY.
***
PLAIN TRIGONOMETRY.
Th
HE buſineſs of trigonometry is to find the angles when
the ſides are given, and the ſides, or ratio of the fides,
when the angles are given ; an: to find fides and angles, when
fides and angles are given. Tor which, it is neceffary, that, not
only the periphery of the vircie, but likewiſe certain right lines
in it, be ſuppoſed divided into ſome determinate number of parts.
The ancient geometers have ſuppored the periphery divided into
300 parts or degrees, and every degree into 60 minutes, and eve-
ry minute into 6o feconds, &c.; and every angle is laid to be of
ſuch a number of degrees and minutes as there are in that part
of the periphery meaſuring the argle.
I.
An arch is any part of the periphery or circumference, and is
the meaſure of the angie at the center which it ſubtends.
II.
The quadrant of a circle is one fourth part of the circumference;
the difference of an arch from a quadrant or go degrees, is
called the complement of that arch.
III.
A chord or ſubtenſe, is a right line drawn from one part of
an arch to another.
IV. The
PLAIN TRIGONOMETRY.
149
2 3. 3
bis, 4
IV.
The right ſine, or ſine of any arch, is a right line drawn from
the vertex of an arch perpendicular to the diameter of the
circle, and is equal to half the chord of double that arch a.
If the arch DB, (fig. for the def.) is an arch of 30 deg. DE is the
fine of 30 deg. and twice DE, equal 10, is the ſubtenſe of 60
deg. The fine of 30 deg. is equal one half radius b.
V.
Every line, as DL, divides the radius into two parts, that part
betwixt the center and line, as CE, is called the coane; and
the part betwixt the line and arch, as EB, is the verſed line
of the arch DB. For the ſame reaſon, AE is the verſed ane
of the arch AD; therefore, the verſed fine may be equal,
greater, or leſs than the radius.
VI.
The arch HD is the complement of BD to a quadrant; and FD,
equal CE, is the fine of that arch or cofine of BD.
VII.
If a right line, BG, is drawn from the point B, at right angles
to the diameter, and meeting the right line CG, pailing thro'
the point D; then BG is the tangent of the arch BD, and CG
is the ſecant of that arch,
The right line HI, drawn from the point H, at right angles to
CH, and meeting CG produced in I, is the tangent of the
arch HD, or cotangent of BD; and CI is the fecant of HD,
or coſecant of BD.
The ſine totus, or greateſt fine, is the radius of the circle, which
is the fine of go deg.
C 15. 3
Characters ufeil. + Addition. - Subtraction. * Multipli-
cation. = Equality. :: Proportion. ♡ Extraction of the
ſquare-root.
\\\
When a line is drawn over any number of quantities, theſe
quantities are to be conſidered as one quantity. The marks
· put overany numbers, are to be read degr. min.
fecond third minutes, &c. as 230, 17, 18", 25', &c. Like-
wiſe, R. fignifies Rad. S. Sine, CofCofine, T. Tang.
Cot. Cotangent, Sec. Secant, and Cofec. Coſecant.
S c H O L I U M.
Becauſe the triangles CED, CBG, (fig. for the definitions,)
are ſimilar, CE: ED :: CB : BG, by alter. CE : CB :: ED:
BG, i. e. Col :R :: Sine: Tangent.
Again,
150
PLAIN TRIGONOMETRY.
d 34. I
ܪ
:
Again, CE: CD :: CB: CG, i. e. Cof. : R.::R.: Secant. And,
becauſe the triangles CDF, CED, CBG, and CHI, are fimi.
lar, CE : ED :: CF : FD; but CE-FD ; therefore ED=CF d;
therefore CE is the line of the angle CDE=DCF.
Again, EC : ED :: CB : BG; altern. EC: CB:ED:BG;
therefore, if EC be nearly equal to CB, ED will be nearly equal
to BG ; therefore, if the arch DB be a very imall arch, the line
and tangent are nearly to one another in the ratio of equa-
lity.
II.
Becauſe the chord of any arch, and its ſupplement to a circle,
is the ſame, ſo the fine, tangent, or ſccant, of any arch, and
its ſupplement to a femicircle, is likewiſe the fame.
Ill.
If two ſides of a right angled triangle be given, the other can
be found by 47. El. 1.
.
PRO P. I.
IN N a right angled triangle, if the hypothenuſe be made radiuse
then the ſides are the fines of their oppofito angles; and if ei-
ther of the ſides about the right angle be mi de radinis, the other ſide
is the tangent of its oppoſite angle, and the hypothenuſe is the fecant
of that angle.
>
a def. 1.
For, in the triangle ABC, if, with the center A, and diſtance
AC, a circle be deſcribed, and B procuced till it cut that
circle in D, (No 1.); then CB is the fine of the arch CD, or of
the angle A9, and AB is the cofine of CD, or line of the
angle C. Again, (No 2.) if AB is made radius, and the arch
BD drawn, then BC is the tangent of the arch BD, or of the
angle A; and AC is the fecant of that angle. Wherefore, &c.
Cor. Hence, as AC, Rad. taken in any given meaſure, is to
BC, taken in the ſame meaſure, are ſo any parts into which the
radius is ſuppoſed to be divided, viz. 10.000000, to a number
exprefſing the parts in proportion to the length of the fine of the
angle, that is,
AC being radius, AC:BC::R:S,A
And
AC:BA::R:S,C
AB Rad.
AB:BC ::R:T,A
And
AB: AC::R:Sec. A.
BC Piad.
BC:BA::R:T,C
T
And
BC:AC::R:Sec. C.
PROP
PLAIN TRIGONOMETRY.
151
A
PRO P.
II.
TH He ſides of plain triangles are to one another as the fines of
their oppoſite angles.
C8
I.
3
d I.
Let ABC be the triangle, ahout which deſcribe a circle
ABC^; from the center D let fall perpendiculars upon each of a S. 4o.
the ſides AB, BC, AC, which wil be biſected in the points
E, F, and Gb; but the angle BDE C is equal to the angle b 3. 3.
CDE, and BE is the fine of the angle BDE d, or of the angle
BAC' ;. For the fame reaſon, BF is the line of the angle ACB, e 21. 1.
and GC the fine of the angle ABC: Therefore, BE is to BF as
twice BE is to twice 5F ; that is, BC is to BA as the fine of the .
angle A is to the one of the angle C.
2u. If the triingle is right angled, then BD, the Rad. is the
fine of the right angle; the other two angles as before.
34. If the triangle is obtuſe angled, then, if a triangle is
formed upon the ſame baſe, in the oppoſite ſegment in the point
I, then that angle will be acute; and BE is the line of the angle
f ſchol. I.
BIC, or BACf. Wherefore, &c.
(
PROP.
III.
I
N any right lined triangle, the ſum of any two ſides, is to their
2:11 rence, as the tangent of half the f1.2 of the angles at the
balı, is to the tangent o half their difference.
Let ABC be the triangle, the ſum of any two of its fides, as
AB, BC, is to the dillerence of theſe fides, as the tangent of
half the ſum of the angics BAC, ACB, at the baſe, is to the
tangent of half their difference.
For, let AB be produced to H; make BH equal to BC ; and
cut off BI equal to BA; then AH is equal to the ſum of the
ſides, and HI to the difference of the fides, and the angle HBC
equal to the ſum of the angles at the baſe?, viz. the angles BAC, a 32. I.
ACB. Join HC; and, from B, let BE fall perpendicular upon
HC b; then, becauſe HB is cqual to BC, the angle BHC is equalb 12. I.
to BCHC, and the angles BEC, BEH, are equall; therefore the c s. 1.
s
angles HBE, EBC, are likewile equald; and, if BE be made rad 32. 1.
dius, then EC is the tingent of half the ſum of the angles at the
baſe. Draw BD parallel to AC, then the angle DBC is equal
to the angle ACB Take HF equal to DC, and join FB;e 29. 1
.
Ia
then
$
>
>
с
+
.
152
PLAIN TRIGONOMETRY.
1 2. 6.
8 14. 5.
then FBD is the difference of the angles, and EBD half their
difference: Through I draw IG parallel to BD or AC; then IB
is to BA as GD is to DCf; but IB is equal to BA; therefore
GD is equal to DC &; but AH is to HC as HI is to HGh; and,
h 4. 6.- by altern. AH is to HI as HC is to HG, the conſequents being
halved, as HA is to HI, ſo is Į HC to HG; but HF, GD, are
each equal to DC, and therefore equal to one another. Add,
or take away, GF to or from both, then HG is equal FD;
but half FD'is ED; therefore AH is to HI as EC is to ED,
Wherefore, &c.
Z
2.
1
PRO P. IV.
THE O R.
N
I N any triangle, the rectangle under half the ſum of the ſides, and
exceſs of the ſame, above any of the ſides, taken as the baſe, is to
the rectangle contained by the right lines, by which the haf of the
Sum of the ſides exceeds the other two ſides, as the square of the
rad. is to the ſquare of the tangent of half the angle oppoſite to the
baſe.
و
244
9
Let ABC be the triangle, BC the baſe ; in the triangle ABC
let a circle be inſcribed a, of which let G be the center, and let
fail GD, GE, GF, perpendiculars to the ſides AB, BC, AC;
then AD is equal to AE, BD to BF, and CE to CF; and the
angles at A and B biſected by the right lines AG, BG ; pro-
duce AB, AC, to H, L; make BH equal to FC, and CL to
BF; at the points H, L, raiſe the perpendiculars HK, LK,
meeting the right line AG, produced in K; from the point K
let fall KM perpendicular to BC; and join BK, KC ; then the
rectangle HAD will be to the rectangle BFC as the ſquare of
AD to the ſquare of DG.
For, the triangles ADG, AEG, are equiangulara, and the
angle ADG equal to the angle AHK, for each are right ones ;
therefore DG is parallel to HKD; and, ſince the angles Alik,
HAK, are equal to the angles KAL, ALK, for AK biſects
them, AH is equal to AL“; but the angles ABC, CBH, are
equal to two right angles d; and DGF, DBF, equal to two right
angles C; forthe angles at D and F are right ones. Take the angle
DBF from both, and there remains DGF equal to HBM ; but
1
HEM, HKM, are equal to two right angles, for the angles at
H and M are right ones; thereforc ihe quadrilateral figures
BDGF, BHKM, are equiangular, and DG bifeds the angles
DBF, DGF ; therefore BK will likewiſe biſect the angles
,
HBM,
b 28, I.
b
c 26. I
d 33. I.
C 23° J.
с
a
PLAIN TRIGONOMETRY153
.
.
3
>
HBM, HKM ; therefore HB will be equal to BM, and the tri-
angle DBG equiangular to BHK. For the ſame reaſon, MCL
.
.
will be biſected by CK, and MC equal to CL.
Now, becauſe BF, FC, are equal to BH, CL; AH, AL,
are equal to the ſum of the ſides AB, BC, AC, and AH equal
to half the ſum of the sides. And, becauſe the triangles DBG,
BHK, are equiangular, GD is to DB as BH is to HK f; and f 4. 6.
the rectangle under DG, HK, equal to the rectangle DBH, 6 16. 6.
that is, to BFC; but the triangles ADG, AHK, are equian-
gular ; therefore AD is to DG as AH is to HK, and the rec-
tangle under DG, HK, equal to the rectangle HAD; there-
fore the rectangle HAD is to the rectangle DBH, or BFC, as
the ſquare of AD is to the ſquare of DG, but AD is the exceſs b 22. 6.
of AH above the baſe BC, and BF, FC the right lines by which
AH exceeds the fides AB, AC; and, if AD is taken rad. then
DG is the tangent of half the angle BAC. Wherefore, &c.
6
PRO P. V. THE O R.
IN.every plain triangle, the baſe is to the ſum of the ſides as the
difference of the ſides is to the ſum or difference of the ſegments
of the baſe, as the greater or leſer ſide of the triangle is taken for
the baſe.
ز
greater fide.
Let ABC he a triangle ; from the vertex A let fall the
perpendicular AD, then the baſe is to the ſum of the fides
as the difference of the ſides is to the ſum or difference of
the ſegments CD, BD, according as the baſe BC is the lefſer or
From the vertex A, let fall the perpendicular
AD upon the baſe BC; with the center A, and diſtance AC,
the greater of the other two fides, deſcribe the circle CEF, and
produce AB both ways to F and E, and CB to G ; then, be- .
cauſe the right lines FE, GC, cut one another in B, the rec-
tangle FBE is equal to the rectangle GBC"; but CB is to BF asa 35. 3.
BE is to GB 0; that is, when the baſe BC is the greateſt, theb 16. O.
baſe to the ſum of the fides, as the difference of the fides to the
difference of the ſegments of the baſe ; but, when BC is the
leaſt, GB is the ſum of the ſegments of the baſe. Wherefore,
Sic.
ܪ
PRO P. VI.
THE fum and difference of any two quantities being given to
find theſe quantities.
U
Let
154
PLAIN TRIGONOMETRY.
*
Let AB, BC, be the two quantities ; place them in the ſame
right line, as AC ; and biſect AC in E ; and cut off AD equal
to BC; then DB is the difference of the two quantities, and EB
half their difference, therefore, if to AE, half their fum, EB,
half their difference, be added, the ſum is equal to AB, the
greater quantity; and if from AE, half the fum, ED, half their
difference, be taken, gives AD equal to BC, the lefſer quantity,
Wherefore, &co
.
THE
1
1
1
2
1
1
PLATN TRIGONOMETRY
Fig for the Definition
H
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Prop. 1.
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TRIGONOMETRY,
本​中
​DEFINITIONS
}
a
A
I.
TH
T HE poles of a ſphere are two points in the fuperficies of
the ſphere that are the extremes of the axis.
II.
The pole of a circle in a ſphere, is a point in the ſuperficies of
the ſphere from which all right lines, drawn to the circum.
ference of the circle, are equal to one another.
III.
A great circle in a ſphere is that whoſe plain paſſes through the
center of the ſphere, and whoſe center is the ſame with that
of the ſphere, or whoſe plain biſects the ſphere.
IV:
A ſpherical triangle, is a figure comprehended under the arches
of three great circles of a ſphere.
V.
A ſpherical angle is that which is contained under two arches of
greater circles in the ſuperficies of the ſphere:
P R O P.
I.
GE
RE AT circles in a sphere mutually bijeet each other.
biſeft
Let the two great circles be ACB, AFB, they will mutually
bifect each other; for their common fection AB is the diameter
of both circles.
PROP:
156
SPHERICAL TRIGONOMETRY.
PROP.
II.
IF from the pole of any circle, to its center, a right line be drawn,
it will be perpendicular to the plain of that circle.
3
a
Let the circle be AFB, and its pole C; from which draw CD
to the center, then CD will be perpendicular to the plain of that
circle.
For, in it draw any diameters EF, GH, and join CG, CH,
CE, CF; then, in the triangles CDF, CDE, the two fides
CD, DE, are equal to the two fides CD, DF, and their bafes
à def. 2.
CF, CE, are equal“; therefore the angle CDF is equal to the
angle CDE D; therefore CD is perpendicular to the plain of the
circle AFB b. Wherefore, &c.
COR. I. Hence, if this circle be a great circle, the diſtance
upon the ſuperficies of the ſphere betwixt the pole and great'
circle is a quadrant, for the plain of it biſects the ſphere.
II. Great circles, that paſs through the pole of ſome other
circle, make right angles with it ; for the right line CD is the
d 19. II. common ſection of ſuch plains d.
b 8. I.
C4. II.
.
P R O P. III.
IE F a great circle is deſcribed about the pole of a fahere, and from
that pole two right lines 'be drawn to the circle, the arch of that
circle contained by the two right lines is the meaſure of the angle
at the pole.
а
a cor. *
Let A be the pole of a ſphere, and ECF the great circle de.
fcribed about it, and let the right lines AC, AF, be drawn to
the great circle; then the arch CF is the meaſure of the angle
at A.
For, let D be the center of the ſphere, then the angles ADC,
ADF, are right angles"; and the angle CDF is the inclination
of the plains ACB, AFB, and equal to the ſpherical triangle
b def. 6. CAF, or CBFb.
COR. I. If the arches AC, AF, are quadrants, then A is the
pole of the circle paſſing through the points C, F; for AD is at
right angles to the plain FDC
II. The vertical angles are equal, for each is equal to the in-
clination of the circles; alſo, the adjacent angles are equal to
two right angles.
II.
و
C4. II.
PROP,
--
SPHERICAL TRIGONOMETRY.
157
1
PRO P.
IV.
3
IF
F two ſpherical triangles have two ſides of the one, equal to two
fides of the other, and the angle contained by the two ſides of
the one equal to the correſpondent angles of the other, the two tri-
angles will be equal.
2
>
For, if the two arches containing the angles are equal, their
chords or ſubtenſes are likewiſe equal", and contain equal angles; a 29. 3
therefore their baſes are equal, and remaining angles of the one,
equal to the remaining angles of the other, each to each; and the
right lined triangles equal b; but equal right lines cut off equal b 4. 1.
circumferences ; wherefore the ſpherical triangles are equal to © 28. 34:*
one another.
COR. I. Hence triangles will be equal and congruous, if two
angles of the one be equal to two angles of the other, each to
each, and a ſide of the one equal to a ſide of the other, either
the ſide that lies betwixt the equal angles, or ſubtending one of
them d.
d 29. and
II. Equilateral triangles are likewiſe equiangular..
24. 3. and
III. In iſoſceles triangles, the angles at the baſes are equal ; e 29. and
and, if the angles at the baſes are equal, the triangles are ifol- 34, 3., and
celes f.
f 29 and
IV. Any two fides of a triangle are greater than the third; for 24. 3. S.
any two of their chords or ſubtenſes, are greater than the third 8.
and 6, 1.
820, 1.
26. 1.
I. I.
a
1
PROP.
V.
Anr
Nir ſide of a ſpherical triangle is leſs than a ſemicircle.
Let AC, AB, the ſides of the triangle ABC, be produced
till they meet in D, then the ſemicircle ACD is greater than
the arch AC.
P R O P.
VI.
*
THE
He three ſides of a spherical triangle are leſs than a whole
circles
For,
158
SPHERICAL TRIGONOMETRY.
I cor. H
For, BD, DC, two ſides of the triangle BCD, are greater
than the third BC 4. Add BA, AC; then DBA, DCA, the
two femicircles, are greater than the three ſides of the triangle
BCD b. Wherefore, &c.
P R O P.
VII.
1
IN any triangle, the greater angle is ſubtended by the greater
fide.
Let ABC be the triangle, and A the greater angle, then BC
will be the greater ſide. For, make the angle BAD equal to the
a cor. 23. angle B; then AD will be equal to BD"; therefore the fide
BDC is equal to AD and DC; but AD, DC, are greater than
ACD; therefore BC is greater than AC. Wherefore, &c.
2. 4
PROP.
VIII.
1
IN
N any Jpherical triangle, if the ſum of two of its fides be great-
er than a ſemicircle, then the internal angle at the baſe will be
greater than the external and oppoſite angle ; and the ſum of the
internal angles at the baſe will be greater than two right angles ;
if equal, equal, and, if lefs, leſs.
3.
Let ABC be the ſpherical triangle ; if the two ſides, AB, BC,
be greater than a ſemicircle, the internal angle BAC, at the
bafe, will be greater than the external and oppofite angle BCD;
if equal, equal, and, if lefs, leſs; and the angles A and
ACB will likewiſe be greater, equal, or leſs, than two right
angles. Firſt, let the ſemicircles ACD, ABD, be compleated;
then, if AB, BC, be equal to ABD, the angle BCD will be
equal to BDC“, that is, to BAC. 2dly, If AB, BC, be great-
er than ABD, then BC will be greater than BD, and the angle
BDC, that is, the angle BAC, greater than BCD b; if AB,
BC, are leſs than a ſemicircle ; then the angle A will be leſs
;
than BCD. And, becauſe the angles BCD, BCA, are equal
to two right angles, if the angle A be greater than the angle
BCD, then the angles A and ACB will be greater than two
right angles, and, if lefs, leſs.
PROP.
) 7.
SPHERICAL TRIGONOMETRY159
NOM
.
1
PRO P.
IX.
IF
F the poles of the ſides of any Spherical triangle be joined by
great circles, they conſtitute another triangle, the ſides of which
are ſupplements of the arches that meaſure the angles of the given
triangles ; and the arches that are the meaſures of the angles of
the ſupplementary triangle, are the ſupplements of the ſides of the
given triangle.
Ler G, H, D, be the angular points of the given triangle
GHD; and let the points G, H, D, be the poles of the great
circles XCAM, TMNO, XKBN; then XN will be the fup.
plement of BK, XM of CA, and MN of OT. Likewiſe, the
arches KT, OC, and BA, which are the meaſures of the angles
M, X, N, are the ſupplements of HD, HG, and GD.
For, becauſe G is the pole of the circle XCAM, GM is a
quadranta ; and, becauſe H is the pole of the circle TMO, HM a cor. 1.2.
is alſo a quadrant : Wherefore, M is the pole of the circle GHb. d cor. 1. 3
For the ſame reaſon, N is the pole of the circle HD, and X the
pole of the circle GD. Now, becauſe NK, XB, are each qua-
drants, XN is the ſupplement of KB. For the ſame reaſon, XM
,
is the ſupplement of AC, and MN of OT ; which are the mea-
ſures of the angles G, H, D.
Again, becauſe DK, HT, are each quadrants, KT is the
ſupplement of HD. For the ſame reaſon, OC is the ſupple-
ment of GH, and BA of GD; that is, the meaſures of the
angles X, M, N, are the ſupplements of the ſides HD, GH,
and GD. Wherefore, &c.
PROP.
X.
1
HE three angles of a ſpherical triangle are greater than two
right angles, and leſs than fix.
i
a
Let the triangle be GHD; then the three meaſures of it,
with the three ſides of the triangle XMN, are equal to three
ſemicircles a; but the three ſides of the triangle X MN are leſs a g.
than two cemicircles b; therefore the meaſure of the three angles b 6.
G, H, D, are greater than one; that is, greater than two right
angles ; but the outward and inward angles of any triangle are
together equal to fix right angles; therefore the inward angles
are leſs than fix right angles. Wherefore, &c.
PRO P.
160
SPHERICAL TRIGONOMETRY.
P R O P
XI.
1
IT, in any great circle
, a point is taken, which is not the pole of
it, and
from that point ſeveral arches are drawn to its circum.
ference, the greateſis of theſe arches is that which palles through
the pole ; and the remainder of it is the leaſt; and the arch nearer
to that, paling through the pole, is greater than that more remote;
and they make obtufe angles with the great circle
II.
b 7. 3.
Let AFBE be a great circle, and any point R taken, which is
not the pole of it, and from that point the arches RA, RB, RG,
RV, of great circles to the circumference of AFBE, the arch
RCA, which paſſes through the pole, is the greateſt, and RB is
the leaſt ; and the arch RCA is greater than RG, and RG
greater than RV.
For, becauſe C is the pole of the circle AFB, CD and RS,
2. and 7. that is parallel to CD, are perpendicular to the plain AFB,
from the point S draw SA, SG, SV; then SA is the greateſt
line, viz. greater than SG, and SG greater than SV b. For,
in the right angled plain triangles RSA, RSG, RSV, the
ſquares of RS, SA, that is, the ſquare of RA, is greater than
the ſquares of RS, SG, that is, than the ſquare of RG, that is,
RA is greater than RG. For the ſame reaſon, RG is greater
than RV; therefore, the arch RA is greater than the arch RG,
and RG greater than RV.
Again, the angle RGA is greater than the angle CGA,
which is a right angle"; and the angle RVA greater than
CVA, a part of it ; therefore the angles RGA, RVA, are ob-
& cor. 3.
с
tuſe angles.
PRO P.
XI.
IF F the ſides containing the right angles of a ſpherical triangle be
of the ſame affection
with the oppoſite angles, that is, if the ſides
are greater or leſs than quadrants, the oppoſite angles will be
greater or leſs than right angles.
Let AGR, AGX, be right angled ſpherical triangles, ha-
ving the angles GÄR, GAX, right ones; then, if the ride
AR be greater than a quadrant,
the angle AGR will be greater
than a right angle; and, if AX be leſs than a quadrant, the
angle AGX is leſs than a right angle.
For,
SPHERICAL TRIGONOMETRY.
161
!
For, if AC is a quadrant, C is the pole of the circle AFB,
and the angles AGC, AVC, are right ones; therefore the ſide
AR fubtending the angle AGR, is greater than a right angle a; a 7.
and; becauſe AX is leſs than a quadrant, the angle AGX is leſs
than a right angle.
a
1
P R O P.
XIII.
IF
F the two ſides containing the angle of a ſpherical triangle be
both leſs, or both greater than quadrants, then the hypothenuſe
is leſs than a quadrant.
In the triangle ARV, or BRV, let F be the pole of the circle
AR; then RF is a quadrant, which is greater than RV.
P R O P.
XIV.
IM
F one of the ſides is greater, and the other leſs than a quadrant,
then the hypothenuſe will be greater than a quadrant.
For, in the triangle ARG, the hypothenuſe RG is greater
than RF, that is, greater than a quadrant. For the ſame rea-
fon, if the hypothen uſe is greater than a quadrant, then one
of the legs is greater, and the other leſs, than a quadrant.
P R OP.
XV.
IE
F the angles at the baſe of a ſpherical triangle he both leſs, or
both greater than quadrants, the perpendicular will fall with
in the triangle ; but, if one be greater, and the other iefs, the per-
pendicular will fall without the triangle.
Let ABC be the triangle; from the point A let fall the per
pendicular AP; in the firſt caſe, it will fall within the triangle;
but, if not, it will fall without; then, in the triangle APB,
the ſide AP, and angle B, are of the fame affection, and like-Fig. *.
wiſe the ſide AP, and angle ACP: Therefore, ſince the angles
ABC, and ACP, are of the ſame affection, the angles ACB
and ABC are of different affections; but they are not. Where-By Hyp.
fore, &c.
x
x
in
i
162
SPHERICAL TRIGONOMETRY.
}
Fig. I.
In the ſecond cafe, if the perpendicular does not fall without,
let it fall within. Then, in the triangle ABP, the angle B,
and fide AP, are of the ſame affection, and likewiſe, in the
triangle ACP, the angle C, and ſide AP, are of the ſame af-
fection; therefore, the angles B and C are of the ſame affection
which is impoffible?. Wherefore, &c.
ܪ
2
а Нур,
P R O P.
XVI.
IN
N right angled ſpherical triangles, having the ſame or equal
acute angles at the baſes the lines of the hypothenuſe are propor-
tional to the fines of the perpendicular arches, and the fines of the
baſes proportional to the tangents of the perpendicular arches.
Let the triangles be BAC, BHE, right angled'at A and H,
and the ſame acute angle B, at the baſe BA, the fines of the hy-
pothenufes CP, CE, are to one another as the fines of the per-
pendicular arches CD, EF; and the fines of the baſis AQ,
HK, proportional to IA, GH, the tangents of the perpendicular
arches.
For, becauſe CD, EF, are perpendicular to the ſame plain,
a 9. 11. 'they are parallela, as alſo, FR, DP; therefore the plains of the
$ 18, 11. triangles EFR, CDP, are parallel b; and CP, ER, the common
fections of theſe plains, with the plains paſſing through BE, EO,
C 16.11. will be parallel “; therefore, the triangles CDP, EFR, are equi-
angular; wherefore CP, the fine of the hypothenuſe BC, is to
cü, the fine of the perpendicular arch CA, as ER, the fine of
the hypothenuſe BE, is to EF, the fine of the perpendicular
d ko 6. arch EH d. For the ſame reaſon, the triangles QAI, KHG,
are equiangular. Wherefore QA is to AI, as KH is to HG 4,
the tangents of the perpendicular arches. Wherefore, &c.
b
9
с
1
PROP.
XVII
|
I
N any right angled Spherical triangle, the cofine of the angle at
the baſe, is to the fine of the vertical angle, as the cofine of the
perpendicular is to the radius.
iſt, Let the triangle be ABC, right angled at A, the cofine
of B is to the fine of the angle ACB, as the cofine of CA is to
radius. For, let the fides AB, BC, CA, be produced, ſo that
BE, BF, CI, CH, be quadrants ; from the poles B, C, draw
the great circles EFDG, IHG; then the angles át E, F, I, H,
و
>
are
SPHERICAL TRIGONOMETRY.
163
a
C
C 16.
are right angles ; therefore D is the pole of BAE', and G the a 2. cor.
pole of IFCB ; AE the complement of the arch AB, and FE e-
qual GD, the meaſure of the angle B, and DF their comple-
ment; and likewiſe BC equal IF, the meaſure of the angle G,
and CF their complement; and CA, one of the fides about the
right angle, equal to HD, and DC their complement.
2d, In the triangles HIC, DCF, right angled at I and F, ha- .
ving the ſame acute angle C. Since BA is leſs than a quadrant,
DF is to DC as IH is to HC b; but HC is a quadrant; there- b 4. 6.
fore, the line of DF is to the fine of DC, as the fine of HI is
to the fine of HC, that is, to radius. Wherefore, &c.
3d, For the ſame reaſon, in the triangles AED, CFD, right-
angled at E, F, and having the ſame acute angle D, as AE is
leſs than a quadrant, the fine of EA is to the fine of CF, as the
fine of LA is ſo the fine of DC, that is, the cofine of the baſe is
to the cofine of the hypothenuſe, fo is radius to the cofiné of the
perpendicular.
4th, Again, in the triangies BAC, BEF; right angled
at A, E, and having the ſame acute angle B; as S. BA is to the
S. BE, ſo is the T. AC to the T. EF; that is, the fine of the
baſe is to the rad. as the tangent of the perpendicular to the tan
gent of the angle at the baſe.
sth. In the triangle GIF, GHD, right angled at I and H,
and having the ſame acute angle G; S. GH is to the S. GI, as
the T. HỒ to the T. IF; that is, the cofine of the vertical angle
C is to rad. as the tangent of the perpendicular is to the tangent
of the hypothenuſe.
6th. Again, in the ſame triangles S. IF is to S. GF, as S. HD
to fine GD; that is, the S. of the hypothenuſe is to rad. as the
fine of the perpendicular is to the fine of the angle at the
baſe.
7th. In the triangles HIC, DFC, right-angled at 1, F, and
the ſame acute angle C, as S. CI is to S. CF, ſo is T. HI to
T. DF, that is, Rad. is to Coſ. BC, ſo is T. C to Cot. B and
S. CH: S. HI. :: S. DC : S. DF ; that is, Rad. is to S. ACB in
Cof. AC : Cof. B.
C
PROP
P R O P.
XVIII,
THE coſines of the angles at the baſe are proportional to the
fines of the vertical angles.
Let BCD be the triangle, either obtuſe or acute angled ; let
{all a perpendicular CA ; then, as the coline of B is to S. BCA,
fo
164
SPHERICAL TRIGONOMETRY.
fo is the cof. CA to radius; and as cof. CA is to radius, fo is car.
D to the S. DCA a, therefore, by equality b, coſ. B is to
S. BCA fo is cof. D to S. DCA.
.
bayi
& 25. 3:
b
P R O P.
XIX.
IN
Nevery Spherical triangle, the cofines of the ſides are propor-
tional to the cojines of the baſes.
1
a 14
Let BCD be the triangle, the ſame things ſuppoſed as in the
laſt, the cof. BC is to the cof. BA as coſ. CA is to rad.; and.
cof CA is torad. as cof. DC is to coſ. DAX; wherefore the cor.
j
BC is to cof. DC as cof. BA is to the cof. of DAb.
a
D 23. 5.
PROP
XX.
IN Noblique ſpherical triangles, the fines of the baſes are in the re-
ciprocal proportion of the tangents of the angle at the baſe.
.
a 7
The ſame things ſuppoſed as before, S. BA is to rad. as
T. AC is to the tangent of the angle at Ba; and, inverſely,
radius is to S. BA as T. B is to T. AC; and S. DA is to rad
as T. D is to. T. AC; wherefore, S. BA is to S. DA as T. D
is to T. Bb.
b 23. $.
PRO P.
XXI.
I N oblique ſpherical triangles, the tangents of the ſides are in the
reciprocal proportion of the cofines of the vertical angles.
a 17
The ſame things ſuppoſed as before, R. is to the cof. ACB as
T. BC to T. AC a, and R. is to cof. ACD as T. DC to T. AC a
and cof. ACD XT. DC is equal to R. X T. AC; therefore cof.
ACB x T. BC is equal to R. X T. AC 6, therefore cof.
ACB X T. BC is equal to cof. ACD XT. DCC; therefore
T.BC is to T. DC, as cof. ACD to cof. ACB d.
b
b 16.6.
сах. І. І.
d 14.6.
P R O - P.
XXII.
INC
N every oblique ſpherical triangle, the fines of the fides are pror
portional to the fines of their oppoſite angles.
For
SPHERICAL TRIGONOMETRY.
165
}
2
For, the ſame things ſuppoſed as before, S. BC is to rad, as a 14.
b 23. So
S. CA is to S. Bà; and S. DC is to rad. as S. AC is to S.D;
and, inverſely, R. is to S. BC as S. D is to S. AC; wherefore,
S. BC is to S. DC as S. D is to S. Bb.
PROP.
XXIII.
I N every Spherical triangle, the cotangent of half the ſum of the
angles at the baſe, is to the tangent of half their difference, as
the tangent of half the vertical angle to the tangent of the angle
that the perpendicular makes with the line bifecting the vertical
angle.
Let BDC be the triangle, and let CF bifect the vertical angle
C; then, as rad. : S. ACB :: cof. AC:cóf. of the angle B , a 176
and rad. : S. ACD :: cof. AC:cof. D; therefore, by eq. and per-
mutation, cof. B : cof. D::S. ACB:S. ACD; therefore, cof. B +
cof. D: coſ. B cof. D :: S. ACB + S. ACD: S. ACB
S. ACD :: cot. B +D:T.B-D::T.BCF:T. ACF=T.BD.
:
2
2
22
PRO P. XXIV.
IN any Spherical triangle, the rectangle contained under the fines
of two ſides, is to the ſquare of the radius, as the difference of
the verſed fines of the baſe, and difference of the ſides, to the verſed
fine of the angle oppoſite to the baſe.
Let ABC be the triangle, and CF, AE, the fines of the fides
AB and CB, or MB = CB; then CF = MF X AE: AO XON
:: IL, the difference of the verfed fines of the baſe AC, and the
difference of the ſides BC and BA, to the verſed ſine of the
angle B.
For, deſcribe a great circle PN about the pole B ; let BP,
BN, be quadrants; then the arch PN is the meaſure of the
angle B. From the ſame pole B deſcribe a leffer circle CFM
through C; the plains of theſe circles will be perpendicular to
the plain BON *; and, let PG, CH, be perpendicular to the
ſame plain, they will fall on the common ſections ON, FM“, a 38. 11,
ſuppoſe in G and H ; and through H draw HI perpendicular to
AO; then the plain paſſing through CH, HI, will be perpen-
dicular to the plain AOB; and AI, which is perpendicular to
HI, is likewiſe perpendicular to CIb; and AI is the verſed fine b 4. II.
of
;
166 SPHERICAL TRIGONOMETRY.
hero
C
:
ܪ
of the arch AC, and AL the verſed fine of the arch AM, which
is equal to BM - BA = BC - BA; and, becauſe MF, NO,
- - ---
€ 16. 11. are parallel to CF, POC, the iſoſceles triangles are equiangu-
lar
; therefore, if perpendiculars CH, PG, be drawn to the
fidcs FM, ON, the triangles will be divided ſimilarly ; there-
fore FM ON :: MH : GN ; and, becauſe the triangles AOE,
DIH, DLM, are equiangular AE : A0 :: IL : MH ; for IL
= ID, DL, and MH = MD, DH; but FM: ON :: MH :
1:3, 6.
GN; therefore d, AE X FM, or CF :: AO X ON : ILX MH:
MH X GN; the two laſt divide by MH, it will be AE X FM :
AO X ON:: IL : GN ; that is, the rectangle under the fines of
the legs is to the ſquare of the rad. as the difference of the ver-
fed fine of the baſe, and the difference of the legs, is to the
verſed ſine of the angle B. Wherefore, &c.
;
d
>
P R O P.
PROP
XXV.
IN any ſpherical triangle, the difference of the verſed fines of
two arches, multiplied into half the radius, is equal to the réc-
tangle contained by the fine of half the fum, and the fine of half the
difference of theſe arche's.
ܪ
Let there be two arches AC, BA, whoſe difference BC let be
bifected in D; then AD is half the ſum, and DB half the dif-
ference of theſe arches; and BT = GH the difference of their
verfed fines, and CE the fine of half their difference. Then,
becauſe the triangles ODV, CBT, are equiangular, as DV:
BT :: OD: CB :: DO : { CB ; wherefore DV XBC ; or,
DV X CE = BT X DO = HG X DO. Wherefore, &c.
2
PRO P.
XXVI.
T
HE verſed fine of any arch, multiplied by half the radius,
is equal to the ſquare of the fine of half of that arch.
Let the arch be DB, and C its center ; draw the right lines
,
DB, BC, and DE, perpendicular to BC ; then DE is the fine
of that arch; which let be biſected by the right line CM; then
;
are the triangles CMB, DEB, equiangular; for the angles at
M and E are right ones; and the angle at B is common; there.
fore EB : BD :: BM : BC; therefore EB X BC = BM BD,
and EB X BC = BMX BD = BM ſquare. Wherefore, &c.
0
ܪ
:
.
PROP
O
SPHERICAL TRIGONOMETRY.
167
-
1
PRO P.
XXVII.
T
IN any ſpherical triangle, the rectangle contained under the fines
of any two ſides, is to the ſquare of the radius, as the rectangle
contained by the fines of theſe arches, which are the baſe and the dif.
ference of the ſides and the fine of that arch, which is half the dif-
ference of the ſame, to the ſquare of the fine of half the angle opo
poſite to the baſe.
Let the triangle be ABC, and BC, BA, the fides containing
the angle B, and AC the bafe fubtending that angle; and lec
;
the arch AM be taken equal to the difference of the legs; then,
the rectangle under the fines of the ſides BC, BA, will be to the
ſquare of the radius, as the rectangle under the line of the arch
AC + AM; and the fine of the arch AC - AM is to the fine
2
2
of one half the angle B.
For, becauſe the
rectangle under the lines of the legs AB, BC,
is to the ſquare of the radius, as IL is to the verſed fine of the
angle B a; and, fince R XIL equal to the rectangle under the a 246
fines of the arches AC + AM, and AC - AM b; and half b 25.
Z
.
2
2
radius multiplied by the verfed fine of the angle B, is equal to
the ſquare of the line of one half the angle B“; therefore the c 26.
rectangle under the fine of the ſides, is to the ſquare of the rad.
as the rectangle under the lines of the arches AC + AM and
AC - AM is to the ſquare of the line of one half the angle B.
Wherefore, &c.
c
2
А
:
+
ray
R
( 168 )
A ſhort EXPLANATION of the TABLES of LOGARITHMS.
and SINES and TANGENTS.
S
UPPOSING now the young geometrician ac-
quainted with the Elements of Euclid, that he may like.
wiſe be acquainted with the Practice, as well as the Elements of
Trigonometry, for which the Tables of Logarithms of Numbers,
of Sines, Tangents, &c. are neceſſary ; but, not being ſo far ad-
;
vanced in mathematics as to underſtand the proper method of
conſtructing theſe tables, that depending in a great meaſure on
infinite ſeries ; although, at the ſame time, as much is contained
in the Elements of Trigonometry as is ſufficient to explain the
nature of theſe, I ſhall only here give a ſhort definition of Lo-
garithms, and ſhow their different uſes in the practice of Trigo-
nometry, &c.
Logarithms, the invention of Lord Napier, publiſhed in the
year 1614, improved by himſelf and Mr Brigs of Oxford Col-
lege are ſuch an arrangement of artificial numbers, with natu.
ral ones, that the addition of the artificial numbers anſwer to
the multiplication of the natural ones, in ſuch a ſeries as is
judged moſt convenient by the calculator. That ſeries which at
preſent is judged moſt commodious is that which increaſes in a
tenfold proportion, as 1; 10, 100, 1000, 10000, &c. the arti-
ficial numbers anſwering to theſe, are, 0, 1, 2, 3, 4, &c. which
are a ſeries of numbers, in arithmetical progreſſion, beginning
with o, as above, which are the exponents or the indices of the
former; the addition or ſubtraction of which indices anſwer to
the multiplication or diviſion of the numbers, as 04=+1+3=2+2,
&c. which are likewiſe the indices of IX10000=10X1000=100
X100, &c. any number of arithmetical means taken betwixt
I and 10, 10 and 100, &c. anſwering to the intermediate na-
tural numbers, in ſuch manner as the addition of the one an-
ſwers to the multiplication of the other, taken to any number of
places, conſtitute a Table of Logarithms, which, taken inſtead
of their correſpondent numbers, many tedious operations are e-
vited; e. g. if two numbers are to be multiplied together, add
their log.; if to be divided, ſubtract the log. of the diviſor from
the log of the dividend ; if a number is to be ſquared, double
its log. ; if to be cubed, multiply its log. by 3; if the ſquare-root
is to be extracted, half its log. ; if the cube root of any number
is to be extracted, take of its log. the number anſwering to
which is the root required; the fame of any other power. In
finding the log, anſwering to any number, if the number con-
Gift of one place, the exponent is o ; if of 2 places, is i;
if of 3 places, 2, &c.; and, if the log, of any number have its
expo.
a
ch
SPECBICAL TRIGONOMETRY
1
}
li
1
1
ERRATA for plane and ſpherical Trigonometry:
Pl. Trig: pr. 4. 1. 18. for c. 23.1 r. c. 32. I.
Spb. Trig. pr. 3. for triangle CAF r. angle CAF.-pr. 6. for 1 cor. 48.
cor. 4.4.- pr. 7. note for a cor. 23. r. a cor. 3.4. for b 2.4. r. b cor. 4.fe
pr. 8. for a 3. r. a cor. 3. 4pr. 11. for c. cor. 3. F. C. 7. p. 164., for a
23.5. r. b 22. 5; the ſame in pr. 19. and pr. 22.
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exponent o, the number is from 1 to 10; if the exponent is 2,
the number is from 10 to 100; if 3, from 100 to 1000, &c,
But, as the following table conſiſts only, of 4 places, or from
I to 10,000, which is for the moſt part ſufficient; but, if not, by
the following eaſy proportion, it may be extended to any num-
ber of places thought neceſſary. If of 5 places, find the firſt four
places, three of which are in the column below N, anıt their log.
in the column next, on the right hand, in the column below o;
and the others below the fourth figure, which will be found a.
long the top of the column, except ſuch figures as are in com-
mon with thoſe in the firſt columns; but, if the fourth figure
to the right hand is a cypher, its log. is the ſame as the three
firſt figures, but the exponent 3 ; the ſame, if all the places to
the right hand of the ſignificant figures are cyphers, only putting
their exponent according the number of places, as before direc-
ted; but, if the figures to the right hand are ſome fignificant fi-
gures, the firſt four being found, find the log, of the next great-
er; then, as that difference is to 1, and cyphers to the number
of places, fo is the figure given to the log. of theſe figures.
Example 1. What is the log. of 47675? The log. of 47670 is
is 4.6782452, the next greater is 47080=4.6783362; then, as
10, the diff. of the numbers, is to 910, the diff. of their log. ſo
is 5, the number wanted, to 455, the log. to be added to the
5
log. 4.6782452 = 4.6782907.
Ex. 2. What is the log. of 47675178? The log. of 47670000 is
7.6782452, next greater, viz. 47680000, is 7.6783362; then;
as 10000, the diff. of the number, is to 910, the diff. of the log.
ſo is 5478 to 448, the log. to be added to 7.6782452 = 7.6782-
950.
Any Logarithm being given, to find the number anſwering to it.
ܪ
Ex. What is the number anſwering to the log. of 7.5571689?
which number, from the exponent, muſt conſiſt of 8 places.
The neareſt log, in the table is .5571461, their difference is
228 ; the difference between the log. found and next greater, is
1204. Then, as 1204 is to 10000, the diff. in places from the
number found in the table, and that wanted ; fo is 228 to
1893.6 ; which, placed to the right hand of the number already
found, is the number anſwering to the given log. thus 360718.
93.6 ; and ſo of
If the log. of a fraction is wanted, ſubtract the log. of the de.
nominator from the log of the numerator ; the remainder is the
log. fought ; but its exponent negative the ſame of a decimal
fraction.
Y
;
any other.
( 190 )
If the log.of a mixed number is wanted, find the log of the whole
number, and th: fraction, as above, whoſe log. add to the log.
of the whole number, or reduce the mixed number to an impro-
per fraction, and find its log. as above. If the log. of a whole
number, and decimal fraction is wanted, find the log. as if all
were whole numbers ; but prefix the exponents only for the
whole numbers. If, at any time, the hyperbolic, or Neper's
log. is wanted, the modulus betwixt which and Brigg's log is
the decimal .434294481,03, &c. Suppole the hyperbolic log.
10 is wanted, divide Brigg's log. of 10=1,000ooco by the above
modulus, gives 2,302585092994; the fame of any other. If the
hyperbolic log. is given, being multiplied by the modulus, gives
Brigg's log.
Of SINES, TANGENTS, &c.
ވާ
THE radius of the circle being ſuppoſed divided into any
rumber of parts, the fines of ſuch a number of theſe parts as
the arches, of which they are the fines, are of a quadrant ; and,
as the tangent of 45° is equal radius, the parts of the tangent a-
bove 45° will be proporționally greater than radius : The ſame
of ſecants, as they are always greater than radius; but the lines
being given, the tangents and fecants can be found from the
proportion given page 150 ; the verſed fine may be found thus,
becauſe the cofine of any arch + the verſed fine, is equal to
rad. Therefore, from rad, ſubtract the cofine, gives the verſed
.
fine required. The above are natural lines, tangents, &c.
The log. ſines, tangents, &c. are only the log. of the natural
fines, tangents, &c. ſo that from the log. tables of numbers,
find the log. anſwering the number of the log. fine, tangent,
fecants, verfed fines, gives the log. fine, tangent, &c. of the
arch required
Ex. If the log. Ene of 500 is required, its natural line is
7660,444, of ſuch parts as the rad. is 1.000,000, the log. of
which fine is 9.8842540 ; and as many places as the natural
fine wants of 1. ſo many units will the log. fine want of index
9; and as many places as the natural tangent or fecant exceeds
units, ſo many units will the log, tangent, or ſecant, exceed in.
dex 10 ; but the log. fine, tangent, &c. may be found without
the log. tables, by an infinite feries, the rad. of the log. being
10.0000000 ; the ſame of any other ſine. If the tangent of any
a fchol. I.
arch, as of 50° is wanted, then, becauſe the tangent of any arch is
pl. trig.
a fourth proportional to the cof. fine, and rad., therefore adding
the log One to rad. and, ſubtracting its coſine, gives the tángent.
of that arch.
Ex.
I
( 171 )
;
Ex. Log S. of 50° + rad.-19.8842540_log. of 40 =9,8080.
675 = 10.0761865 = tang. 50°; and, becauſe the fecant is a
third proportional to the cofine and rad. therefore, from twice
rad. ſubtract the cofine of 50°, gives 10.1919325, 'the ſecant of
50°. If the verſed ſine is wanred, ſuppoſe of 50, ſubtract its
cofine from rad. gives 3572.124, its natural fine; its log. fine,
found, as before directed, is 9.5529265: The ſame thing of
any other. But, becauſe the verſed line of an arch above 90° iş
frequently wanted, the exceſs of the arch above rad. + rad. =
the verſed fine above 90°, Suppoſe the verſed line of 130° is
wanted, its exceſs above rad. is 40°, the natural fine of which is
6427.876 + R. =16.27.876, its log, is 10.2155814; or, with-
out the natural fines, thus, the log. of 2 = 30103000 + twice
the log. fine of half the arch, = 20.21558.4 --- rad. = 10.2155-
.
814 = log. verſed fine of 130º.
In the table of fines, tangents, &c. begin, as uſual, in all tables
of the like nature with one minute, increaſing to 45°, the degrees
below 45° are placed on the left ſide of the page; the minutes
anſwering to them increaſe downwards; thoſe above 45° are
placed on the right ſide of the page; the minutes anſwering to
them increate from the bottom of the page upwards, as uſual.
3
To find the nat. or log. fine, tangent, &c. of ſecond and third Mi-
nutes, for which a Table is calculated, for every ſecond ; and
each 6" to 5 places, at the end of Table of fines, tangents, &c.
5
30"
TO find which, find the degrees and minutes in the table of
fines, tangents, &c. and take the difference betwixt that iound and
- next greater; then, as i'is to that d fference, ſo is ſecond and
d
third minutes, required in the denon ination of a minute, which
is done in decimals, in the table, at the end of the tables of ſines,
&c. to a fourth proportional; which add to the line or tangent
found in the table of fines, &c.
Ex. If the nat. fine of 580, 19, 22", 361", is wanted, find the
,
ſine of 589.19', the difference betwixt which and 589.20', the next
greater is 1,528; which, x the decimal of 22", 36", as found
in the table, gives 576.056 +8509, 639 = 8510,215, the deci.
mal.o56 being rejected. If the log. fine of 58° 19', 22', 36'", is
wanted, find log. ſine 58° 19', the difference betwixt which and
58° 20'is 779 x 3:7, found, as before, is 293.683 + log. ſine of
58°19' = 9.9299112 = 9.9299405, the decimal 683 being rejected,
as before : The fame of any other fine, tangent, fecant, whether
above or below 45°, or of any verſed fine, whether above or bea
>
low 90°
If
( 172 )
--
If ani arch is given, to find the degrees, minutes, fecordss and
thirds anſwering to that arch.
I
ލް
>
Find in the table of fines, &c. the neareſt, and take the dif-
ference betwixt that found and arch given ; divide that dif-
ference by the difference betwixt that found and next greater,
gives a quotient; which, in the table at the end of the fines,
&c. ſhows the ſecond and third minutes anſwering to it.
Ex. Let the arch given be a fine of 9.9876543, the neareſt
leſs is 9.9876488, the line of 70°.24', the difference is 55;
which divide by the difference betwixt the line of 76°, 24, and
76° 25' = 306, gives -17973, which look for in the table of fe-
conds and thirds, gives 10", 45" Obſerve the ſame of any other
arch, whether tangent, fecant, or verſed fine.
a
1
7
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T A B
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L O G A R I T H M
OG A RI T Н M S
For NUMBER s increaſing in their Natural Order,
from an Unite to 10,000,
With a Table of Artificial SINES, TANGENTS,
and SECANTS, the Radius 10,0000000; and to
every Degree and Minute of the Quadrant,
Num.
Log.
Num.
Log.
Num.
Log.
І
34
35
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36
37
38
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0.0000000
0.3010300
0.4771213
0.6020600
0.6989700
0.7781513
0.8450980
0.9030900
0.9542425
1.0000000
1.0413927
1.0791812
I.1139434
1.1461280
1.1760913
I.2041200
1.2304489
1.2552725
1.2787536
1.3010300
1.3222193
I.3424227
1.3617278
1.3802112
1.3979400
I 4149733
1.4313638
1.4471580
1.4623980
1.4771213
1.4913617
1.5051500
1.5185139
42
43
44
45
46
47
48
49
so
SI
52
53
54
55
56
57
$8
59
60
61
62
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
1.5314789
1.5440680
1.5563025
1.5682017
1.5797836
1.5910646
1.6020600
1.6127839
1.6232493
1.6334685
1.6434527
1.6532125
1.6627578
1.6720979
1,6812412
1.6901961
1.6989700
1.7075702
1.7160033
1.7242759
1.7323938
1.7403627
1.7481880
1.7558749
1.7634280
1.7708520
1.7781513
1.7853298
1.7923917
1.7993405
1.8061800
1.8129134
1.8195439
А
O O O O OO O OO
COV O WNHO
82
83
84
85
86
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1.8260748
1.8325089
1.8388491
1.8450980
1.8'512583
1.8573325
1.8633229
1.8692317
1.8750613
1.8808136
1.8864907
1.8920946
1.8976271
1.9030900.
1.9084850
1.9138139
1.9190781
1.9242793
1.9294189
1.9344985
1.9395193
1.9444827
1 9493900
1.9542425
1.9590414
1.9637878
1.9684829
1.9731279
1.9777236
1.9822712
1.9867717
1.9912262
1.9956352
lo e na
88
89
90
91
92
93
94
95
96
97
98
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63
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65
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9
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100
IOI
000000004441 08677 1300917337 21661 25980 30295 | 34605 38912
0043214 47512 51805 56094 603806466068937 1 73210 7747881742
1020086002 | 90257945099875603000 107239 11474 15704 19931 | 24154
103 | 0128372 32587 36797 4100345205 49403 53598 5778861974 66155
1040170333 | 74507 178677 82843 | 87005 0116395317 i 9946703613 07755
105 0211893 16027 20157 24284 28406 32525 36639 | 4075044857 48960
1060253059 57154 | 61245 65333 69416 73496177572 $1644 8571389777
1070293838 9789501948 05997 10043 14085 1812322157 26188 36214
108 0334238-38257 42273 4628550293 54297 58298 62295 66289 70219
|
|
|
109 0374265 78248 | 82226 | 86202 90173 94141 98106 02066 06023 09977
110 0413927 | 1787321816 25755 29691 3362337551 | 41476 45398 | 49315
|
III 0453230 57141 610486495268852 72749766428053284418 88301
112 04921809605699929 03798 07663 11525 15384 1923923091 26939
113 0530784 34626 38464 42299 46131 4995953783 57605 61423 65237
114 | 0569049 728567666180462 842608805591846 95634 99419 03200
|
115 0606978 10753 14525 18293 220582582029578 33334 / 37086 40834
]
|
1161 0644580 48322 52061 | 55797 59530 | 6325966986 70709 7442878145
|
117 1 0681859 85569 8927692980 966810037904073 07765 1145315138
118 0718820 22499 26175 29847 133517 37184 40847 445074816451819
119 0755470 5911862763 66404 70043 | 73679 177312 8094284568 88192
120079181295430 69045 02656 06265 0987013473 17073 20669 24263
1210827854 3144135026 | 38608 42187 | 45763 49336 52906 5647360037
122 | 0863598 6715770712 74265 77814 181361 84905 8844691984 95519
123 0899051 02581 06107 | 09631 13152 16670 20185 2369.71 27206 30713
124 0934217 37718 41216 44711 48204 $1694 55180 58665 ) 6214665624
125 0969100 72573 76043 79511 | 82975 | 864378989693353 96806 | 00257
|
126 1003705 07151 10594 14034 1747120905 | 24337 | 2776631193 34616
127 1038037414564487148284151694 55102 5850761909 65309 68705
128 1072100 75491 78880 82267 856508903192410 95785 99159 02529.
| |
129 110589709262 12625 15985 19343 22698 26050 2940032747 36092
130 113943442773 46110 49444 52776 56105 59432 62756 7607769396
131 1172713 76027 79338 82647 859548925892559 9585899154 02448
132 1205739 09028 12315 1559818880 2215925435 2870931981 35250
133 1238516 41781 45042 48301 5155854813 58065613146456167806
134 1271048 74288 77525 80760 83993 1 87223 90451 93676 96899 00119
|
135 | 1303338 06553 09767 12978 16187 1939322597 25798 28998 32195
136 1335389 38581 41771 | 44959 48144 51327 54507 | 57685 60861 64034
137 136720670375 | 73541 76705 | 79867 | 83027 86184 89339 92492 | 95643
|
138 1398791 0193705080 0822211361 14498 17632 20765 23895 27022
139 1430148 3327136392 39511 42628 4574248854 51964 55072 58177
|
140 1461280 64381 67480 70577 | 73671 | 76763 79853 / 82941 | 86027 | 89110
141 14921919527098347 0142204494 07564 10633 13699 16762 | 19824
142 | 1522883 25941 2899632049 35100 38149 | 41195 4424047282 50322
143 1553360 $6396 594306246265492 168519 171544 74568 2758980608
144 1583625866408965392663 9567298678 01683 104685 07686 10684
|
145 1613680 16674 19666 2265625644 28630 | 31614 34596 37575 40553
| |
| |
146 | 1643529 4650249474 52443 554115837661340 64301 6726170218
147 1673173 76127 79078 82027 84975 879209086493805 9674499682
148 1702617 05551 08482 | 1141214339 1726520188 23110 2602928947
149 1731863 3477637688 40598 43506 46412 49316 52218 55118 58016
| /
150 1760913 63807 66699 69590 72478 | 75365 78250 | 81133 84013 86892
ISI 1789769 | 92645 955182838901259 041260699209856 12718 15598
152 1818436 21292 24147 26999 298503269835545 38390 41234 44075
| |
153 1846914 497525258855422 58254 6108463912 66739 69563 | 72386
154 1875207 78026 80844 8365986473 89285 | 92095 94903 9771000514
155 1903317 061180891711715 14510 1730420096 22886 25675 28461
156| 1931246 34029 3681039590 42367 45143 47918 50690 53461 56229
157 195899761762 64525 67287 70047 | 72806 75562 78317 81070 83821
158 198657189319 92065 94809' 97552 00293 03032 05769 08505' 11239
,
159) 2013971 16702 | 19431 22158 24883127607 130329 33049 35768) 38485
|
4
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|-
2
7
5
3
4
6
160 2041200 43913 46625 / 49335 52044 54750 57455 | 60159 62860 65560
1612068259 70955 7365076344 79035 | 81725 84414 87100 89785 92468
|
162 2095150 9783000508 03185 05860 08534 11 205 1387616544 19211
163 2121876 24540 27202 29862 3252135178 37833 4048743139 45790
164 2148438 51086 53732 56376 5901861659 6429866936 69572 72207
165 2174839774718010082729 8539587980 90603 | 93225958.45 98464
166 2201081 036960631008922 11533 14942 16750 1935621960 24563
167 2227165 29764 32363 | 34959 | 37555 40148 42740 453314792050507
| /
|
168 2253093 5567758260 60841 6342165999 68576 71151 73724 76296
169 227886781436 84004 86570 | 89134 91697942589681899377 01934
|
1702304489 07043 09596 12146 14696 17244 1979022335 24879 27421
|
1712329961 32500 350381 37574 40108 42641 45173 47703 50232 52759
172 2355284 5780960331 | 62853 65373 6789170408 72923 75437 77950
|
173 2380461 82971 85479 | 87986 90491 | 92995 95497 9799800498 02996
|
174 2405492 07988 10482 12974 15465 17954 | 20442 22929 | 2541427898
175 2430380 32861 35341 37819 40296 4277145245 47718 50189 52658
176 2455127 57594 60059 62523 64986 67447 69907 7236574823 77278
177 2479733 82186 | 84637 | 87087 8953691984 94430 96874 99318 01759
/
/
178 250420006639 09077 11513 13949 16382 18815 21246 23675 26103
179 2528530 30956 33380 35803 3822440645 43063 45481 47897 50317
180 2552725 55137 57548 59957 62365 | 64772 67177 0958271984 74386
1812576786791858158283978 86373 88766 91158 93549 9593798327
182 | 2600714 03099 05484 07867 10248 12629 15008 17385 19762 22137
183 2624511 26883 29255 31625 33993 36361 38727 41092 43455 45817
|
184 2648178 50538 52896 55253 57609 59964623176466967020 69369
185 2671717 74064 76410 78754 / 81097 8343985780 88119 9045792794
186 2695129 97464 99797 0212904459 06788 09116 11443 13769 16093
187 2718416 2073823058 25378 27696 30013 32328 34643 3695639268
/
188274157843888 46196 48503 50809 53114 55417 57719 60020 62320
189 2764618 66915 69211 71506 73800 7609278383 80673 829621 85250
|
190278753689821 92105 94388 96669 9895001229 03507 05784 08059
191 2810334 12607 14879 17150 19419 21688 23955 262312848630750
192 283301235274 37534 39793 42051 4430746563 48817) 51070 53327
193 2855573 57823 6007102319 64565 66810 69054 71296 73538 75778
|
194 287801780255 | 82492 8472886963 89196 91428 93660 9589098118
|
195 290034602573 04798 0702209246 11468 13689 15908 18127 20344
196 2922561 24776 2699029203 31415 33626) 35835 38044 40251 42457
|
197 2944662 4686649069 51271 53471 5567157869 60067 6226364458
198 296665268845 71037 73227 75417| 77605 79792 81979 | 84164 86348
199 2988531 90713 9289395073 9725299429 0160503781 05955 08128
200 3010300 12471 14641 16809 18977 2114423309 25474 27637 29799
201 3031961 34121 3628038438|40595 4275144905 47059 49212 31363
202 3053514 55663 57812 5995962105 64250 6639468537 | 7068072820
203 307496077099 79237 81374 83509 85644 87778 89910 92042 94172
|
|
204 3096302 | 98430 00557 02684 04809 06933 0905611178 13300 15420
205 3117539 19657 21774 23889 26004 28118 30231 | 32343 34454 36563
206 313867240780 42887 44992 47097 49201 51303 534055550557605
207 315970361801 63898 65993 68088 7018172273 | 74365 7645578545
208 3180633 8272184807 86893 8897791061 93143 95224 97305 99384
209 3201463 0354005617 07692 09767 11840 13913 15984 18055 20124
210 3222193 24261 26327 28393 30457 3252134584 36645 38706 40766
| )
211 3242825 44882 46939 4899551050 5310455157 57209 59260 01310
|
212 3263359 65407 67454 6950071545 73589 | 75633 2767579716 81757
213 328379685834 87872 899099194493979 9601298045 00077 02108
214 3304138 00167 08195 10222 | 1224814273 1629718320 20343 22364
215 3324385 26404 28423 30440 32457 34473/ 36488 38501 40514 42526
216 334453846548 4855750565 52573 54579 56585 5858960593 62596
217 3364597 66598 68598 70597 72595 74593 7658978584 80579 82572
à 3384565 | 8655788547 9053792526 94514 9650 l '18301 | 20277 | 22252
| |
1
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9
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8
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7
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​6
3
4
220 3424227 | 26200 2817330145 32116 34086 36055 38023-39991 41957
2213443923 45887 47851 49814 5177653737 5569857657 59615 61573
222 3463530 65486 | 6744169395 71348 73300 75252 77202 7915281101
223 3483049 8499686942 888879083292775 94718 96660 98601 00541
224 3502480 04419 06356 | 08293 1022912163 1409816031 17963 19895
225 3521825 23755 25684 27612 2953931465 33391 35376 37239 39162
226 3541084 43006 4492646846 48764 5068252599 54515 56431 58345
227 3560259 6217164038 65994 67905169814 71723 73630 7553777443
228 3579348 81253 8315685059 86961 88862 90762 92662 94560 96458
229 3598355 00251 0214604041 0593407827 09719 11610 13500 15390
236 | 3617278 19166 21053 22939 24825 | 26709 28593 30476 32358 34239
231 3636120 37999 39878 41756 43633 7551047386 49260 51134 53007
232 3654880 56751 58622 60492 62361 64230 66097167964 69830171695
233 3673559 75423 77285 79147 8100882869 84728 8658788445 90302
234 3692759 940149586997723 9957001428 03280 081310698108830
235 3710679 12526 14373 16219 180651990921753 2359625438 27279
236 | 372912030960 32799 34637 36475 38311 4014741983 43817 45651
2373747483 49316 51147 52977 5480756636 58464 60292 6211863944
238 376577067594 694187124073062 74884 -767047852480343 82101
23937839798579687612 89427 91241 93055948689668098492 00302
240 3802112 03922 05730 07538 09345 1115112956 14761 16565 18368
241 3820176 21972 23773 25573 27373 29171 8309932766 34563 36359
242 383815439948 41741 43534 45326 47117 4890850698 52487 54275
243 3836063 578506963661421 63206 6499066773 68555 70337 72118
244 3873898 7567877457 79235 810128278984565 86340 88114 89888
245 | 3891661 93433 9520596975 98746 10051502284|04052 05819 07585
246 3909351 1111612880 14644 10407 18169| 1993121691 23452 2521F
247 392697028727 30485 32241 33997 35752 3750639260 41013 42765
248 3944517 46268 48018 49767 51516 5326455011 56758 58504 60249
349 3961993 6373765480 67223 68964 70705 724467418575924 77663
250 3979400 81137 | 82873 84608 86343 88077 8981191543 93275 95007
251 3996737 10846100196 01925 036530538010710608832 10557 12287
252 4014005 (15728 1745119173 20893 22614 24333 26052 27771 29488
253 4031205 32921 34637 36352 38066 3978041492 43205 44916 46627
254 4048337 50047 51755 53464 55171 5687858584 60289 61994 63698
253 4065402 67105 68807 70508 72209 73909 75608 77307 19005 80703
256 408240084096 8579187486 89180 90874925679425995950 97641
2574099331 01021 0271004398 06085 07772 09459 11144 | 12829 14513
258 4116197 1788019562 21244 22925 2460526285 27964 : 29643 31320
259 4132998 34674 36350 38025 39700 41374 4304744719 : 46391 48063
260 4149733 51404 53073 54742 5641058077 5974461410 63076 64741
261 4166405 68069 69732 71394 73056 74717 7637978037 } 79696 1 81355
2624183013 846708632787983 89638 9129392947 94601 96254 97906
263 | 4199557 01208 02859 04509 0615807806 | 09454 IIIOI12748 14394
264 4216039 17684 19328 20972 22614 24257 25898 27539 29180 30820
265 4232459 34097 35735 37372 ! 39009 40645 42281 43916 ; 45550 47183
2664248816 50449 52081 53712 5534256972 58601 6023061858|63486
267 4265113 66739 6836569990 71614 73238 74861 76484 78106 79727
268 4281348 82968 84588 86207 87825 89443 91060 92677 9429395908
269 4297523 99137 0075102363 039760558807199 08809 10419 12029
2704313638 15246 1685318460 20067 21673 23278 24883 2648728090
2914329693 31295 32897 34498 3609837698 39298 40896 42495 44092
272 4345689 47285 48881 50476 52071 53665 5525956851 58444 60035
293 4361626 63217 64807 66396 67985 695731 71161 72748 74334 75920
274 4377506 7909080675 | 822588384185423 87005885879016791747
2754393327 9490696484 9806299639 012160279204368 105943 07517
276 4409091 10664 12237 13809 15380 16951 1852220092 21661 23230
277 4424798 26365 127932 29499 31065 3263034195 35759 37322 38885
278 4440448 42010 ( 43571 45132 46692 4825249811 51370 52928 54485
2794456042 157598 1591541 60709 1 622646381865372166925 168477 70029
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836 9222063 22582 23102 23621 24140 2465925179 2569826217 26736
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843 9258276 58791 59306 59821 60336 60851 01366 61880 62395 62910
844 9263424 63939 64453 6496865482 65997665116702567539 68053
8459268567 6908169595 70109 7062271136 7165072163 72671 73190
|
846 9273704 74217 74730 75243 75757 76270 26783 77296 77808 78321
847 9278834 79347 79859 80372 80885 81397 81909 82422 82934 83446
|
| |
8489283959 84471 84983 85495 86007 86518 870308754288054 88565
|
849 9289077 89588 9010090611 91123 9103492145 92656 93167 93678
|
850 9294189 94700 9521195722 96233 96743 97254 9776498275 98785
851 9299296 99806 0031600826 01336 01847 02357 02866 163376 63886
337603886
8529304396 04906 0541505925 06434 06944 07453 07963 08472 08981
853930949009999 10508 11017 11526 12035 12544 1305313562 14070
|
854 9314579150871559616104 16612 57121 17629 18137 18645 19153
|
855 1 9319661 20169 20697 21185 21692 2220022708 2321523723 24230
856 9324738 25245 25752 26259 26767) 29274 27781 28288 28795 29301
857 9329898 30315 30822 31328 31835 32341 32848 3335433860 34367
$589334873 353793588536391 36897 37403 37909 38415 38920 39426
859 933993240437 40943 41448 41953 42459 42964 43469 43974 44479
860 9344985 45489 45994 46499 47004 47509 48013 48518 49023 49529
861 9350032 5053651040 51544 5204952553 5305753561 54065 54569
8629355073 555765608056584 57087 57591 58095 58598 59101 59605
863 9360108 6061161114 61617 62120 62623 6312663029 64132 64635
|
8649365137 65640 66143 66645 67148 07650 68152 08655169157 69659
865 9370161 70663 711657566772169 72671 73172 73674 7417674677
|
866 9375179 75680 76182 76683 77184 77686 178187 786887918979690
867 19380191 80692 81193 81693 82194 82695 83195 8369684196 84697
868 9385197 | 85698 86198 866988719887698 88198 88698 89198189698
8699390198 9069791197 | 91692 92196 | 92696 9,3195 93695 94194 04693
87019395193 95092 96191 966909718997688 9818798685 99184 99683
|
891 9400182 00680161179 0169717217002674 03172 03670 04169 04667
892 9405165 05663 06161 06659 0715707654 08152 08650 09147 09645
893 | 9410142 10640 11137 11635 12132 | 12629 13126 13623 14120 14617
874 9415114 15611 16108 | 16605 19101 17598 18095 18591 19088 19584
|
875 9420081 20577 21073 21569 22065 22562 23058 23553 24049 24545
896 94250412553726032 26528 27024 27519 28015 2851012900529501
|
847 9429996 30491 30986 31481 31976 32471 32966 33461 33956134450
/
|
898 9434945 35440 3593436429 36923 37418 37912 | 38406 3890039395
879' 9439889 40383 1 40877141371 141865 ) 4235842852 433464384044333
|
1
No
1
8 i
6
13 14 15
7
1
|
1
I
880 9444827 | 45320 45814 46307 46800 47294 77787 48280 48773 49266
8819449759 50252 50745 51238 517305222552716 53208 53701 54193
882 945466655178 ! 53671 56163 5665557147 57639 58131 58623 59115
8839459607 60099 60591 61082 6157002060 62557 63049 635 40 6403 I
|
8849464523 65014 165505 65996 6640766978 67469 07960 68451 68942
8859469433 69923 70414 70905 71395 71886 72376 72366 / 73357 1 73847
|
8809474337 7482715317 175807 76297 76787177277 77767 78257 178747
887 9479236 79726 80215 1 80705 81194 81684| 2173 82662 83151 | 83641
888 9481130 84619 85108 85597 86085 86574 87063 87552 88040 88529
889 9489018 89506 1 89995 90483 909719146091948 92436 9292493412
890949390094388 94876 195364 95852 9633) 96327 197315 97802 98290
8919498777 99264 99752 00239 : 00726 01213 01701 0218802675 03162
8929503649 04135 04622051091 05596106082 06569 07055 | 07542 08028
8939508515 0900109487 09973 | 10459 10946 11432 11918 12404 12889
894 | 9513375 13861 14347 14832 15318 15803 16289 16774 17260 17745
i
895 9518230 18716 19201 19686 201712065021141 21026; 22111 22595
896952308023565 24049 24534 25018 25503 25987 26472 26956 27440
|
2695627440
8979527924 28409 | 28893 29377 2986130345 30828 | 31312 31796 32280
898 9532763 33247 33731 3421434697! 35101 35664 36147 36631 37114
|
|
8999537597 38080 38563 39046 39529 40012 40404 40.)771 41460 41943
9009542425 42908 43390 4387344355 i 44837 45319 4580246284 46766
901 9547248 47730 48212 40694 | 49176.47657 50133 50321 S1102 51584
902 9552065 5254753028 53510 53491 54472 54953 155434 55916 56397
903 9556878 57358 57839 58320' 5oeci 59282 59762 1602.? | 60723 61204
9049501684 62165 62645 63125 : 63606 64066 04506 65040 65526 66006
905 9566486 66966 67445 67925: 68405 688856.9304 by044 70323 | 70803
9069571282 71761 72241 72720 7319373'707415774636 75115 75594
907 957607376552 7703077509'77988: 78400 70745 | 79423 79902 180380
908 9580858 81337 81815 82273 8277183249 83727 2420534083 | 85161
909 9585639 86117 86594 87072 87549 88027 88505 106982 8945989937
910 9590414 90891 91368 91845 : 92332 92800 73-76 93753 94230 94707
911 9595184 95660 96137 196614 97090 97507 76043 98520 9899699472
912 9599948 00425 00901 | 01377 01853 0232y 32805 03281163756 04232
913 9004708 05183 03659 C-135 00610, 07086 09501 0003608512 08987
9149609462 09937 10412 10887 : 11362 11837) 12312 12707 13262 13736
915 9614211 14686 15160 1563516109 16503 17058 17532 18006 1848r
9169018955 19429 19903 20377: 20851 21325 21799 22272 22746 23220
917 9623693 24167 24640 25114 2558726061 26334 27007 | 27481 27954
9.189628427 28900 29373 29246 130319 | 30792 31204 31737/32210 32683
919 0633155 33628 34100 34573 | 35045 | 35517 3599030402 36934 37406
920 9637878 38350 38822 39294 39766 40238 40710 41181 41653 42125
921 9642596 43068 43539 44011 44482 44953 45425 4589646367 46838
9222647309 47780 48251 48722 49193 49064 50135 5060551076 151546
923 9652017 52488 52958 53428 53899 5436954839 5530955780 50250
924 96567205719057660 58130 58599 159069159539 0c009 6047860948
925 9661417 61887 62356 162826 63295 03764) 64233 647036517265641
926 966611066579 167048 6751767985 68454 6892369392 69860170329
927 9670797 71266 171734 72203 726717313973607-407674544 75012
928 967548075948 76416 76884 77351 77819, 782877475779222 79690
929 9680157 80625 81092 31559 82027 8247482961 33+28 83895 84362
930 | 9684829 85296 85763 86230 86697 187164 876278097 88504 39030
|
9319689497 89963 90430 90896 91302 91829 92295 0270103-2729193
932 9694159 94625 95091 95557 40023 96488 : 26954 97420 97805 90351
!
933 9698816 9928299747 (00213 00678 01143 arcod 0207402539 03004
:
1934 9703469 03934 0439904803 05328 05793 002; 8 06722 0718; 07652
935 9708116 08581 09045 09509 09:14 10430 | 1090 11366 11830 12294
|
o9y
|
936 9712758 13222 · 13686 14156 14614 15078 15$42 16005 1646 16932
(:
937 9917396 17859 18333 18756 19249 19713 : 20176 20639 21102 21565
938 9922028 22491 22954 ; 2341723880-4343 : 244805 25268 25731 26193
939 1 9726656) 27118 27581 28043 28506 48968 |29430 | 29892 1 30354 30816
I
Ι
1
No
a
1
6
2
5
3
4
1
940 9731279 31741 32202 32664 33126 3358334050 34511 3497335435
941 9733896;6358 36819 37281377,42 3820838664/39126 39587 40048
942 974050410970 | 41431 41892 42353 428144327443735 44196 44656
|
943 9745117 +5577 46038 46498 4695947419 47879 48340 48800 49260
944 97497205018050640 51100 51500 52020 5247952939 5339953858
945 19754318 54778 55237 55697 5615656615 5707557534 5799358452
946 975891159370 59829 60288 60749 0120661665 02124 62582 163041
947 976350063958 64417 64875 653346579266251 66709 67167 16,625
:
;
9489768083 68541 69000 69458 69915 70373 17083171289 71747 72204
949 9772662 73120 73577 74035 7449274950 75407 75864 1 76322 76779
950 977723677693 78150 ( 78607 7906479521 79978 | 80435 1 8089281348
951 9781805 8226282718 83175 183631 84088 | 84544 8500185457 85913
|
|
952 9786369 8682687282 87738 88194 38650 89106 89562 90017 90473
953 979092991385 2184022296 92751 93207 9366294118 94573 95028
1
954 9795484 95934 96394 9684997304 9775998214 98669 99124 99579
955 9800034 00483 00943 01398 01852 02307 0276103216 03670 04125
956 9804579 0503305487 05942 06396 | 0685007304 077580821208666
9579809119 09573 10027 1048r 10934 113881184112295 12748 13202
958 9813655 14108 14562 15015 15468 159211637416827 17280 17733
| | |
959 19818186 18639 19092 19544 19997 20450 20902 21355 2180722260
|
960 1982271223165 | 23617 24069 24522 249742542625878 26330 26782
961 9827234 27686 2813828589 29041 29493 29945 30396 30848 31299
|
962 9831751 32202 32654 33105 33556 34007 34459 3491035361 35812
963 983626336714 37165 37616 38066 38517 3896839419 39869 40320
964 9840770412214107142122 42572 4302243473 43923 44373 44823
965 198452734572346173 46623 47073 47523 47973 48422 48872 49322
966 9849771 502215067051120 51569 52019 52468 5291753366 53816
967 19854265 54714 55163 5561256061 56510 56959 57407 57856 58305
968 9858754 59202 5965116009960548 609966144561893 62341 62790
969 1986323863686 04134 64582 6503005498 6592606374 66822167270
970 9867717 68165 | 6861369060 6950869955 704037085071298 71745
|
971 19872192 726407308773534 73981 7442774877 75322 75769 76216
9720876663 77109 7755678003784507889079343 7978980236 80682
973 9881128 81575 82021 | 8246782913 83360 83806 8425284698 85144
74 988559086035 | 86481 86927 87373 8781688264 88710; 89155 89601
|
975 9890046 90492 90937 91382 91828 92273 2271893163 93608 94053
976 9894498 94943953889583396278967220716797612 98057 08501
977 9898946 99390 99835100279 00723 vol 1006120205602500 32944
978990338903833 0429704721 0516405608 06052 064960694007383
979 99078270827108714 0915809601 10044 10438 10931 11374 11818
|
980 9912261 12704 13147 13590 140331447014919 15362 15805 16247
|
981 9916690 17133 17575 180181846118903 19345 1978820230 20673
982992111521557 21999 2244122884 23326 23768 24210 24651 125093
(
983 9925535 25977 26419 26860 27302 27744 28185 28627 29068 29510
984 9929951 30392 30834 31275 31716321573259833039 33480 33921
985 9934362 34803 / 35244 35685 | 36123 36566 3700737447 37888 38329
|
986 9938769 39209 39650 140090 40531 4097141411 41851 42291 +2731
987 19943171 43611 44051 44491 44931 4537145811 4625046690 47130
988 9947569 48009 4844848888 49327 49767502065064551085 51524
989 995196352402528415328053719 54158 5459755036 55474 55913
990 9956352 56791 57229 57668 58106 58545 58983 59422 59860 00298
991 9960736 61175 61613 62051 62489 6292763365 63803 6424144679
|
992 9965117 65554 65992 66430 6686767305 67743 68180 68618 69055
993 9969492 6493070367 170804 71242 716797211672553 72990 73427
994 19973864 743017473875174 75611 76048 7648.76921 77358 77794
|
995 9978231 78667 79104 79540 7997680413 8084981285 81721 82157
996 9982593 83029 83465 83901 84337 84773 8520585645 86080 86516
|
997 9986952 87387 / 8782388258 | 88694 89129 89564 90000 20435 90870
998 9991305 91740 92176 92611 93046 93481 93916 94350 94785 95220
999 9995655 96089 1 96524 196959 I 97393 | 9782898262 98697/ 99131 99566
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Artificial SiNES, TANGENTS, and SECANTS..
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{
18
A Table of Artificial Sines,
M
M
Sine.
o Degree.
Tang.
Secant.
60
I
2.
ороль, но
TI
1 #TAT
I2
I
20
21
30
31
32
6.0000000 10.000000 0.0000000 Infinite.
10.0000000 Infinite.
637261 9.9999999 6.4637261 13.5362739 0000000 13.5362739 59
7647561
9999999 7647562 2352438 0000OOI 235243958
3 9408473
9999998
9408475
0591525 0000002 059152752
4 7.0657860 9999997 | 7.0657863 12.9342137 0000003 12.934214056
5 1626960
9999995 1626964 8373036 0000005
8373040 55
6 2418771
9999993 2418778 7581222
0000007 758122954
7
3088239 9999999
3088248 6911752 0000009 6911761 53
8
36681571 9999988 3668169 6331831 0000012 6331843 52
9 41796811 9999985 4179696
5820304 0000015 5820319 51
IO
4637255 9999982 4637273 5362727 0000018
5362745 50
5051181
9999978 5051203 4948797 0000022 4948819 49
5429065 9999974 5429091 4570909 0000026
4570935 48
13 5776684 9999969 5776715 4223285 0000031 422331647
14 6098530 9999964 6098566 3901434 0000036
3901470 46
IS
6398160 9999959
6398201 3601799 0000041 3601840 | 45
16 6678445
9999953
6678492
3321508
0000047 3321555 44
17 6941733 9999947 6941786 305 8214
0000053
3058267 43
18
189966 9999940
7190026 2809974 0000060 2810034 42
19 7424775 9999934 7424841
2575159
0000066
257522541
7647537 9999927 7647610
2352390 0000073 2352463 40
7859427 9999919 7859508 2140492 0000081 2140573 39
22
8061458 9999911
8061547 1938453 0000089 1938542 38
23 8254507
9999903 8254604 1745396
0000097 1745493 37
27 84393389999894 8439444
1560556 0000106 1560662 36
25
8616623 9999885
8616738 1383262
0000IIS
1383377 35
26 8786953 9999876 8787077 1212923
0000124 1213047 34
27
8950854 9999866 8950988 1049012
0000134
104914633
28 9108793 9999856 9108938 0891062
0000144
0891207 | 32
29 9261190 9999845 9261344
0738656
0000155
0738810 31
9408419 9999835 9408584 0591416
0000165 059158130
9550819 9999823
9550996
0449004
0000177 044918129
9688698 9999812 9688886
Ω3ΙΙΙΙ4 0000188
0311302 28
33
9822334 9999800 9822534
0177466
0000200 0177666 27
34 99519801 9999788 9952192 0047808
0000212 004802026
0077867 9999775 8.0078092 11.9921908 0000225 11.9922133 25
0200207 9999762 0200445 9799555 0000238 9799793 24
0319195
0319446
9680554 0000252
9680805 | 23
0435009 9999735 0435274 9564726 0000265
9564991 22
39 0547814
9999721 0548094
0000279 9452186 21
0657763
0658057
9.341943 0000294 934223720
0764997 9999691 0765306 9234694 0000309 923500319
0869646 9999674 0869970 9130030 0000324 9130354 18
43 0971832 9929660 0972172 9027828 0000340 902816817
44 1071669 9999644 1072025 8927975 0000356 8928331 16
45 1169262 9999628 1169634 8830366 0000372
883073815
1264710 9999611 1265099 8734901 0000389
873529014
47 1358104 9999594 1358510 8641490 0000406
8641896|13
48
1449532 9999577 1449956
8550044 0000423. 8550468 12
49 1539075 9999559 1539516
8460484 0000441 8460925 II
50 1626808
9999541 1627267 8372733 0000459 8373192 10
SI 1712804 9999522 1713282
82.86718 0000478 8287196
9
52 1797129 9999503 1797626
8202374 0000497
8202871 8
53 1879848 | 9999484 1880364 8119636
0000516
8120152
54 1961020 9999464 1961556 8038444 0000536
8038980
55
2040703 9999444 2041259 7958741 OC00556 7959297 5
2118949 9999424 2119526 7880474 0000576 7881051
4
57 2195811 9999403 2196408 7803592 0000597 7804189
3
58 2271335 9999382
2271953 7728047 0000618 7728665
59
2345568 9999360
2346208 7653792 0000640 7654432
60
2418553
9999338
2419215 7580785 0000662 7581447)
8.
35
36
37
38
9999748
945ICOS
9999706
40
41
42
46
WOW CCIO
$6
2
T
Tangents, and Secants.
19
I
9999316
2
736957657
916756654
IO
6729837 47
14
6611437
6612471| 45
6554957 44
6442165 | 42
19.
20
*
24
6068992 35
6018207 34
1 Degree.
M Sine.
Tang.
Secant.
08.2418553 19.99993388.2419215 ( 11.7580785 10.0000662 [ 11.7581447 60
2490332
2491015
0000684
7508985
7509668
59
2560943 9999294 2561649 7438351 0000706 7439057 58
3
2630424
9999271
2631153 7368847 0000729
2698810
2699563
9999247
4
7300437 0000753 730119056
2766136 9999224
5
2766912 7233088 0000776
7233864 55
6 2832434
2833234
9999200
7166766 0000800
2897734 9999175
7
2898559 7101441
0000825 7102266
53
8 2962067 9999150 2962917 7037083
0000850 7037933 52
3025460 9999125
3026335
9
6973665 0000875 6974540 51
3087941
9999100 3088842 6911158 0000900
691205950
3149536
II
9999074 3150462 6849538 0000926 6850464
49
3210269
12
9999047 3211221
6788779 0000953
678973148
13 3270163 9999021 3271143
6728857 0000979
3329243 9998994 3330249
6669751
0001006
6670757 46
15 3389529 9998966 3388563
0001034
16
3445043 9998939 3446105
6553895
OCO1061
17 3501805 9998911 3502895 6497105 0001089
6498195 43
18 3557835 9998882 3558953 6441047
0001118
3613150
3614297 6385703 OOO1147
638685041
3667769
9998824
3668945 6331055
0001176
6332231 40
AI
9998794
3721710
3722915
0001206
6277085
6278290 39
3774988
224
9998764 3776223
6233777
OC01236
6225012 38
3827620
23
9998734 3828886
0001266
6171114
6172380 37
3879622 9998703 3880918 6119082 0001297
6120378 36
25
3931008 9998672 3932336
6067664 0001328
26 3981793
9998641
3983152
6016848 0001359
27 4031990 9998609 4033381 5966619 0001391
5968010 33
4081614 9998577
4083037
5916963 0001423 5918386 32
29
4130676
9998544 4132132 5867868 0001456
5869324 31
30 4179190 9999512 4180679 5819321 0001488
31 4227168 9998478 4228690 5771310 OOO1522
5792832 | 29
4274621 2998445 4276176 5723824 0001555 5725379 28
33 4321561 9998411 4323150 5676850 C001589
34 4367999 9998376 4369622 5630378 0001624 5632001 26
35 4413944 9998342 4415603 5584397 0001658
5586056 25
36 4459409 9998306 4461103 5538897 0001694 554059124
37 4504402 9998271 4506131 5493869 0001729 5495598 23
38 4548934 9998235 4556699 5449301 0001765 $45106622
39 4593013 9998199 4594814
0001801 54069871 21
40 4636649 9998162 4638486 5361514 0001838 $33335120
41
4679850 9998125
4681725 5318275 QO01875 532015019
42. 4722626 9998088 4724538
5275462
0001912 527737418
43
4764984
9998050
4766933 5233067 OO01950 523501017
44 4806932
9998012
4808920 5191080 0001988 $193068 16
45
4848479 9997974
4850505 5149495
0002026
5151521 15
46
4889632 9997935 4891696 5108304
0002065
SI1036814
47 4930398
4932502 5067498 0002104
506900213
4970784
4972928 5027072 COO2144 5029216 12
49 50r0798 9997817
5012982
4987018 0002183
4989202 II
50 5050447 9997776 5052671
4947329 0002224 4949553 IO
5089736
5092001 4987999
0002264 4910204
52 5128673
9997695 5130978
4869022 0002305 4871327
53 5167264 9997653 5169610
4830390
0002347 4832736
54 5205514 9997612
$ 207902 4792098
0002388 4794486
55 5243430 9997570
5245360
4754140 0002430 4756570
5281017 9997527
5283490 4716510 0002473 4718983
57 5318281 9997484
5320797
46792031
0002516
4681719
5355228 9997441 5357787 46422131 0002559 4644772
59 5391863 9997398 5394466
0002602 4608137
bo 5428192 9997354
5430838
4569162 0002646 4571808
28
5820810 30
32
5678439 | 27
5405186
9997896
9997856
48
51
9997730
56
ao routes les Ò
58
4605534
A Table of Artificial Sines,
Secant.
ooou aen te who
II
a to
20
2 2 Degrees.
M Sine.
Tang.
0 8.5428192 19.9997354. 8.5430838 111.456916210.0002646 11.4571808 60
5464:18
9997309 5466909 4533091 0002691 453573259
5499948
9997265 5502083
4497317 0002735 4500052 58
5535386
9997220 5538166
4461834 0002780 4464614 57
5570536
9997174 5573362
4426638 0032826 4429464 :56
5605404
9997128
5608276 4391724
0002872 439459655
5639994
9997082
5642912
4357088 0002918
436000654
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9997036
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5708357
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5742129 9996942 5745197 4254803 0003058
425786151
IO
5775660
9996894 5778766 4221234 0003106 4224340 | 50
5808923 9996846 5812077
4187923
0003154 419107749
12 5841933
9996798 5845136 4154864 0003202 415806748
33 5874694 9996749 5877945 4122055 0003251 4125306 47
14 5907209 9996700
5910509 4089491 C003300 4092791 46
15
5939483 9996650 5942832 4057168 0003350
4060517 45
16
5971517 9996601
5974917 4025083 0003399
4028483 44
17 6003317 9996550
6006767
3993233 0003450 3996683 43
18 6034886 9996500 6038386 3961614 0003500 3965114) 42
19
6066226 9996449
6009777 3930223 0003551 393377441
6097341 9996398 6100943 3899057 0003602 390265940
21
6120235
9996346
6131889 386111 0003654 3871765 39
22
6158910 9996294
6102615 3837384 coo3706 3841090 38
23 6189369
9996242 6193127 3806873
0003758
3810631 | 37
24 6219616
9996189
6223427 3776573 0003811 3780384 36.
25
6249653
9996136
6253518
3746482 0003864 3750347 35
26 6279484
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6283402 3716598 0003918 3720516 34.
27
6309111
9996028
6313083 3686917 0003972
3690889 33
28
0338537
999.5974
6342563 3657437 0004026 3661463 / 32
29
6367764 9995919 6371845 3628155 0004081 3632236 31
39
6395796
9995865
6400931 3599069 0004135 3603204 30
31
6423634
9995809
6429825 3570175 0004191
3574366 29
32 6454282 9995753 6458528 3541472 0004247
3545718 28
33
6402742
9995697
6487044
3512956
0004303
351725827
34
6511016 9995041
6515375
3484625
0004359
3488984 26
35
6539107
9995584
6543522 3456478 0004416
346089325
36
6567017 9995527 6571490 3428510 0004473 3432983 24
6594748
9995469
6599279 3400721 0004531 3405252 23
38
6622303
9995411
6626891
3373109 0004589 337769722
39
6649684 9995353
6654331
3345669
0004647 3350316 21.
6676893
9995295
3318402 0004705 3323107 20
6703932
9995236
6708697 3291303 0004764 3296068
19
42
6730804
9995176 6735628 3264372 0004824 3269196 18
43 6757510
6762393
3237607 0004884 3242490 17
44
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3211004 0004944 3215948'16
45
6810433
9994996
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3189567 15.
6836654
9994935
3158281
0005065
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6862718
9994874
6867844
3132156 0005126
3137282 13
6868625
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6893813 3106187
0005188
3111375 12
49
6914379
9994750 6919629 3080371 0005250
3085621 II
50 6939980
6945292
305*7708 0005312 3060020 10
SI 6965431
6970806 3029194 0005375 3034569
6990734
52
6996172 3003828 0005438
%
3009266
53 7015889
7021390 2978610 COO5502 2984111
7
54 7040899 9994435 7046465 2953535 0005565 2959101 6.
55 7065766 9994370 7071395 2928605 0005630 2934.234 5
7090490
7096185 2903815 0005694 2909510 4
7115075 9994241
7120834 2879166 0005 759
2884925
3
7139520
7145345
2854655 0005824 2860480
2
7163829 9994110
7169719
2830281 0005890 2836171
00 2188002 9994044 2193958
2806042 0005956
2811998 1 0
8 N N N دا ا دا کج کر دی بر ع کر دی پی در پی دی در
37
1
6681598
40
41
9995116
9995056
6815437
6841719
46
47
48
9994688
9994625
9994562
9994498
56
9994306
57
00
58
59
9994176
na trh
I
Tangents, and Secants.
21
3 Degrees.
El
I
OOV Que No head o
9993776
9993703
Ιο
II
A 111
9992938
1
9992198
9992046
M
Sine,
Tang-
Secant.
98.7188002, 9.9994044 87193958 11.2806042 10.0005956, 11.2811998 60
7.21 2040 9993978 7218063 2781937
coo6022 2787960 59
7235946 9993911 7242035 2757965 0906089 2764054.158
3
7259721
999384.4
7265877 2734123 СообI56 274027957
4 7283366
7289589 2710411 0006224 271663456
5
7306882
7313174
2686826 особ292 269311855
6
7330272 9993640 7336631
2663369
Oc06360 2669728 54
7
7353535 9993572 7359964
2046465 53
2640036 0006428
8
7376675 9993503 7383172 2616828 0006497 2623325 52
9
7399691 9993433 7406258 2593742 ooo6567 2609209 51
*7422586 9993364 7429222 2570778
0006636 257741450
7445360 9993293 7452067
2547933
0006707 2554640 49
I 2
7468015 9993223 7474792
2525208
0006777 2531925 48
13
7490553 9993152 7497400 2502600
0006848 2509447 47
14
7512973
9993081 7519892 2480108
0006919 248702746
15
7535278 9993009 7542269 2457731 0006991
2464722 | 45
16
7557469
7564531 2435469 0007062 2442531 43
17
7579546 9992865 7586681
2413319 0007135 2420454 43
18
7601512 9992793
7608719 2391281 0007207
2398488 42
19 7623366 9992720
7630647 2369353
0007280
2376634 41
20
7645III
9992646 7652465 2347535 0007354 2354889 40
21
7666747
9992572
7674175 2325825 CO07428 2333253 39
22
7688275 9992498
7695777 2304223 0007502 2311725 i 38
23 7709697 9992424 7717274 2282726 0007576 229030337
24 7731014 9992349 7738665. 2261335 c007631 2268986 36
25
7752226
9992274 17759952 2240048 0007726
2247774 35
26
7773334
7781130 2218864 0007802 2226666
34
27
7794340 9992122 7902218 2197782 0007878 2205660
33
28
7815244
7,823199 2176801 0007954 218475632
29 7836048 9991969 7844079 2155921 0008031 216395231
30 7856753
9991892 7864861 2135139
0008108 2143247 | 30
31
7877359
9991815
7885544 2114456 coo8185
32 7897867 9991737 7906130 2093870 0008263 2102133 28
33
7918278 9991659 7920620 2073380
0008341
34 7938594
7947014 2052986
9008420
2061406 26
35
7958814 9991501
7967313 2032687 0008499
36
7978941 9991422 7987519 2012481 0008578 202105924
37 7998974 9991342 8007632 1992368 0008653 200102623
38
8018915! 9991262
8027653 1972347 0008738
1981085 127
39 8038764
9991182
8047583 1952417
0008818 196123621
40
8058523
9991101
8067422
1932578 oco8899 1941477 20
8078192
9991020
1912828 0008980
1921808 19
42
8097772
8106834
1893166 co09062 1902228 18
43 8117264
9990356
8126407
1873593
0009144
44 8136668 9990774
8145294 1854106 0007226 186333216
45 8155985 9990691
8165294
1834706 0009309
46
8175217
9990608
8184608 1815392
0009392
1824.783 14
47 8194363 9990525 8203838 1796162
0009475
48
8213425
9990441 8222984
1777016
0009559
1786575 12
49 8232494 9990357
8242046 1757954 0009643
1767596 11
50
8251299
9990273
8261026 1738974 CC09727 174870110
8270112 9990188
8279924 1720076
0009812
1729888
52
8288844
9990103
8298741 1701259 0009897 1711156
53 8307495
9990017 8317478 1682522
0009983 1692505
54 8326066 9989931 8336134 1663866 0010069 16739346
SS 8344557 9989845 8354712 1645288 0010155 1655443
5.
56
8362969
9989758
8373211 1626789 0010242 1637031
57 8381304
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5$ 8399561
9989584
8409977
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8417741
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8428245 1571755 CO10504
15822591
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2122641 29
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was
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8087172
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1882736 177
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SI
axoo no nt man
3
160043911.%.
59
1)
A Table of Artificial Sines,
2
4 Degrees.
M
Sine.
Tang.
Secant.
Paloma
2
OoN ah A
14
9986888
ܐ
o 8.84358451 9.9989408 8.8446437| 11.155356 10.0010592 ; II.1564155 60
8453874 9989319 8.464554 1535440
0010681 154632659
8471827 9989230 8482597 15174.03 OO10770 1528173 58
3
8489707
9989141 8500566 1499434 0010859 I51029357
8507512 9989052 8518461
1481539
0010948 149248856
5
8525245
9988962
8536283
1463717
0011038 147475555
6
8542905
9988871
8554034
1445966 OO11129 1457095 54
7
8560493
9988780
8571713 1428287 COI1220 1439507 53
8 8578010 9988689 8589321 1410679 OO11311 142199052
9
8595457
9988598
8606859
1393141 OOI1402 1.40454351
JO
8612833
9988506
8624327
1375673 0011494
138716750
II
8630139
୨988414
8641725 1358275 0011586 136986149
I 2
8647376 9988321 8659055 1340945 0011679 1352624 48
13
8664545 9988228
8676317 1323683 0011772 133545547
8681646 9988135 8693511 1306489 0011865 131835446
15
8698680 9988041 8710638 1289362 0011959 1301320 | 45
16 8715646 9987947 8727699 1272301 O012053 1284354 44
17 8732546 9987853 8744694 1255306 2012147
126745443
18 874.9381 9987758 8761623
1238377 OOI 2242 1250619 42
19
8766150 9987663 8778487 1221513 0012337 123385041
20 8782854 9987567 8795286 1204714 0012433 1217146 | 40
21 8799493 9987471 8812022 1187978 QO12529 1200507 39
22
8816069 9987375 8828694 II71306 0012625 118393I38
23
8832581 9987278
8845303 1154697 0012722
1167419 37
24
8849031 9987181 8861850 1138150 0012819 115096936
25
8865418
9987084
8878334 1121666 0012916 113458235
26 8881743
9986986
8894757 II05243 0073014
I11825734
27
8898007
8911119 1088881
0013112 I10199333
28 8914209 9986790 8927420 1072580
0013210
108579132
29
8930351 9986691 8943660
1056340 0013309 1069649 31
30
8946433 9986591 8959842 1040158 0013402 1053567 30
31
8962455 9986492 8975962
1024037
0013508
103754529
32
8978418 9986392 8992026 1007974
0013608 1021582 28
33 8994322 9986292 9008030 0991970 0013708
100567827
34
9010168 9986191 9023977 0976023 0013809 0989832 26
35 9025955 99860901 9039866 0960134 0013910 0974045 25
9041685
9985988 9055697 0944303 OO14012 0958315 24
9057358 9985886
9071472 0928528 0014114 0942642 23
9072975 9985784 9087190 0912810 0014216 092702522
9088535
39
9985682 9102853
0097147
0014318 091146521
0881540
9985579 9118400
0014421
40
089596120
0865988
9985475 9134012
41
0014525
088051319
9134881
9985372 9149509 0850491
42
0014623 0865119 18
9150219 9985268 9164952 083504.8 0014732
43
0849781 17
9165504 9985163 0180340 0819000 0014837
44
083449616
9180734 9985058
0804325
99850589195675
45
0014942 0819206 15
46 9193911 9984953 9210957 0789043 0015047 0804089 14
9211034 9984848
47
9226186
0773814 0015152
0788966 13.
9226105 9984742
9241363
0758637
48
0015258 077389512
9241123 9984636
0743513
49
0015364 0758877 II
9256089
50
9984529
0728440 0015471 0743911 | 10
2271003
0713419
99844229286581
SI
0015578
0728997
9
9285866 9984315 9301552 o608448
52
0015685
0714134
8
9300678 99842079316471
0683529 0015793
53
0699322 7
9984099
0668060
9315439
9331340
54
0684561
0015401
6
9330150
9983990
9346160
0653840
55
0016010 0669850 S,
9983881 i 9360929 0639071 0016119 0655189
9359422 99837729375650 0624350 0016228
0640578 3
9390321
0609679 0016337 0626017 20
9388496 9983553
9404944 0595056 0016447
59
0611504 I
94029001 9983442' 9419518 0580482 0016558
0597040
36
37
38
as ba tw w ew
O aathie
911993; 3985579
9256487
9271560
IO
1
9344811
A.
56
57
58
le ta ar
93739839983663
Tangents, and Secants.
23
M
I
2
4
O Vaerew
IO
II
14
0367455
21
24
5 Degrees.
M Sine.
Tang-
Secant.
08.9402960 19.99834428.9419518, 11.0580482 10.0016558 ! 11.05970406.
9417376 9983332 9434044 0565956
0016668 0582624 52
9431743 9983220 9448523 0551477
0016780 056825758
9446063 9983109 9462954 0537046 0016891 055393757
9460335 9982997
9477338 0522662
0017003 053966556
5
9474561
9982885 9491676
0508324 0017115 052543955
6 9488739
9982772 9505967 0494033 0017228 0511261 54
7
9502871 9982660 9520211 0479789 0017340 0497129153
8
9516957 9982546 9534410
0465590 0017454
0483043 52
9
9530996 9982433 9548564 0451436
0017567 046900451
9544991 9982318 9562672 0437328 0017682 045500950
9558940 9982204 9576735 0423265 0017796 044106049
I 2 9572843 9982089 9590754 0409246
0017911
04271571 48
13 9586703 9981974 9604728 0395272
0018026 0413297 47
9600517 9981859 9618659 0381341 0018141
0399483 46
15 2614288 9981743 9632545
0018257 038571245
16 9628014 9981626 9646388 0353612
0018374 0371986 44
17 9641697
9981510 9660188 0339812
C018490 0358303 43
18 9655337 9981393 9673944 0326056 0018607 034466342
19 9668934 9981275 9687658 0312342 0018725 0331066 41
20 9682487
9981158 9701330
0298670 0018842
0317513 40
9695999 9981040 9714959
0285041 0018960 0304001 39
22
9709468 9980921 9728547 0271453 0019079 0290532 38
23
9722895 9980802
9742092 0257908 0019198 027710537
9736280 9980683 9755597 0244403 0019317 026372036
25
9749624 9980563 9769060 0230940 0019437
0290376 35
26 9762926 9980443 9782483
0217517 0019557 0237074 34
27
9776188
9980323
9795865
0204135 0019677 0223812 33
28 9789408 9980202 9809206 0190794 0019798 0210592 32
29 9802589
9980081 9822507 0177493 0019919 0197411 31
30
9815729
9835769
0020040
0184271 30
9828829
9979838
9848991 OIS1009
0020162 0171171 29
32 9841889
9862173 0137827 0020284 QIS8111 28
0158111
33 9854910 9979593 9875317 0124683 0020407 0145090 27
34
9867891 9979470 9888421 OII1579
0020530
0132109 26
9880834 9979347 9901487
0098513 0020653 0119166: 25
36
989.3737
9979223 9914514 0085486 0020777
0066263' 24
37 9906602 9979099 9927503 0072497 0020901
0093398, 23
38 9919429
9978975 9940454 0059546
0021025
0080571 22
39 9932217
9978850
9953367
0046633 0021150
0067783 21
40 9944968 9978725 9966243 0033757 0021275 005503220
41 9957681 9978599 9979081 C020919 0021401 0042319 19
42 9970356 9978473 999188 0008117 0021527 0029644 18
43
9982994 9978347 9.0004647 10.9995353
CO21653
0017006 17
44
9995595
9978220
0017375
9982625 0021780 C004405 | 16
459.0008160
0030000 9969934 0021907 10.9991840 15
46
0020687
9977906
0042721 9957279 C022034 9979313/14
47 0033179
9977838
0055340
9944660
CO22162 9906821 13.
48
0045034 9977710
0067924
CO22290 9954366 12
49
0058053
9977582
0080471
9919522 0022418
9941947 II
50
0070436 9977453 0092984
9907016
0022547 992956410
0082784 9977323 0105461 9894539 0022677
9977216
9
52
0095096 9977194 O117903 988209% 0022806
9904904
53 0107374
0130310 9869690 0022936 9892626 7
54
0119616
୨976933 0142682 9857318 0023067 988038416
55
0131823
9976803
0155021 9844979 0023197
9868177
5
56
0143996
9976672
0167325 9832675 0023328 9856004
4
57
0156135
9976540
0179594
9820406 0023460 98438653
58
0168239
0191831 9808169 0023592
9831761
2
59
0180309
9976276
0204033
9795967
0023724
9819691
60 0192346 9976143
0216202
97037981
0023857 9807654
9979960
0164231
31
9979716
35
>
9978093
9932076
SI
9977064
0
9976408
I
24
A Table of Artificial Sines,
6 Degrees.
Tang.
M
Sine.
Secant.
lo
9975609
97,23448
9974660
12
9974386
9974248
9974110
9973971
9973833
9973693
9581866
18
20
9547164
21
9972850
9972708
9972566
9972423
9972280
9972137
9478561
9455926
9444651
09.019234619-99761439 0216202 10.9783798 10.0023857 / 10.9807654 60
I 0204348 9976011
0228338 9771662
0023989 979565259
2 0216318
9975877
0240441 9759559 0024123 9783682 38
3
0228254
9975 743 0252510
9747490 0024257 9791746 57
4 0240157
0264548
9735452 0024391 975984356
5 0252027 9975475 0276552
6
0024525
0263865
974797355
9975340
0288524
9711476
0024660 973613554
7 0275669
9975205
0300464
9699536
0024795
8
0287442
972433153
9975069
0312373 9687627 0024931
971255852
9 0299182 9974933 0324249 9675751 0025067 970081851
Το
0310890 9974797 0336093 9663907 0025203 968911050
II 0322567
0347906 9652094 0025340 9677433 49
0334212 9974523
0359688 9640312
0025477 966578848
13
0345825
0371439
9628561
0025614 9654175 47
14
0357407
0383159
9616841
0025752 9642593 46
I5 0368958
0394848
9605152 0025890
16
9631042 45
0380477
0406506
9593494 0026029 9619523 | 44
17
391966
0418134
C020167 9608034 43
0403424
0429731
9570269
0026307 959657642
19 0414852
9973554
0441299
9558701
0026446 9585148 41
0426249 9973414 0452836
0026586
957375140
0437617
9973273
0464343
9535657
0026727 9562383 39
22 0448954
9973132
0475821
9524179
0026868
23
9551046138
0460261
9972991
0487270
9512730 0027009 9539739 37
24 0471538
0498689
9501311
0027150 9528462 36
25 0482786
0510078 9489922 0027292
26
9517214 35
0494005
0521439
0027434
27
9505995 | 34
0505194
0532771
9467229 0027577
28
9494806 | 33
O$16354
0544074
0027720 948364632
29 0527485
0555349
0027863
30
9472515 31
0538588
9971993
0566595
9433405
0028007 9461412 30
31
0549661
0577813
9422187
0028151
32
9450339 29
0560706
9971704
0589002
0028296 9439294
28
33
0571723
9971559
0600164
9399836
0028441 9428277 27
34 0582711
9971414
0611297
9388703 0028586
9417289 26
35
0593672
0622403
9377597 0028732 9406328 25
36
0604604
0633482
9366518
0028878
9395396 24
37
0615509
0644533
9355467
0029024 9384491 23
38
0626386
0655556
9344444 0029171 9373614 | 22
39 0637235
0666553
9333447 0029318
9362765 21
40 0648057
9970535
0677522
0029465
9351943 | 20
41
0658852
0688465
9311535 0029613
9341148 19
42
0669619 9970239 0699381 9300619 0029761
9330381 18
43 0680360
0710270 9289730 0029910 931964017
44 0691074
9969941
0721133
0030059
9308926 16
45
0701761 9969792 0731969
9268031 0030208
46
9298239 15
0712421
9969642
Q42779
9257221 0030358 9287579 14
47
0723055 9969492 0753563
9246437 0030508 9276945 13
48
0733663
9969342
0764321 9235679 0030658 9266337 12
49
0744244 9969191
0775053 9224947 0030809
925575611
0754799
9969040 0785760
9214240 0030960
51
0765329
9245201 | 10
0796441
9203559 0031112 9234671 9
52 0775832
0807096
9.192904
0031264 9224168 8
53 0786310
0817726
9182274 0031416
9213690
7
54 0796762
996€431
0828331
9171669
0031569
55 0307189
0838911 9161089
0031722 9192811 5
56 0817590
9968125
0849466
9150534
0031875 9182410 4
57 0827966 9967971
0859996
9140004
0032029
58
9172034 | 3
0838317
9967817
0870501
9129499 0032183 91616833 2
59 0848643
0880981
9119019
0032338
60
9151357
I
0858945 / 996.7507 0891438 9108362
0032493
9141055
9991849
9410998
9971268
9971122
9970976
9970829
9970682
9322478
9970387
9970090
9278867
50
9968888
9968736
9968584
9968278
9203238 6
9967662
$
)
25
Tangents, and Secants.
ing Degrees.
1
2
00 acm A
095.3667
TO
IC
12
13
20
21
22
8884492
82 Degrees.
M
Sine.
Tang
Secart.
09.0858945) 9.9967507 9.0891438 10.9108562 10 0032493 10.9141055, 60
I 08692219967353 0901869 9098131 00326431 9130779 52
0879473 9967196 0912271 9087723 0032804
9120527 158
3
0889700
9967040 0922660 9077340 0032960 9110300 57
0899903
9966884 0933020 9565980 0033116
9100097 56
0910092 9966727 0943355 9056645 0033273 9089918 55
6 0920237 9966579
9046333 0033430 9079763 54.
7
0932367 9966412
09h3955
9036045 0033588 906463353
8 0940474 9966254 0974219 9025781 0033740
9059526 52
୨
0950556 99660:6 0984460 9015540 0033904
9049144 51
0960515 9965937 0994678 9005322 C034063 9039335 150
0970651 9965778 1004872 8995128 0034222 9029.349 149
0980062 9965619 IOI5044 8983956 0034381 9059.335 to
0990651 9965459 10251)2
8974808
0034541 C.934947
14
1000616 9965299 1035317 8964683 0034701 8999384 46
IS 10105581 996;138 104.5 420 8954520 0034862 8901442 145
IÓ
1020477
9964977, :
1055500 8944500 0035023 8979523 44
17 103,373 9964816
1065557 8934443 0035184 8969627 4.3
18 10402461 9964655 1075591 8924409 0035345 8959754 / 42
19 1050096 9964493 1085604 8914396 C035507 8949904 41
1059924 9964330 1095504 8904406 0035670 8940076 40
1069729 9964167 1105562 8894438 0035833 803027139
1079512 9964004 ILIS508
0035996 892048838
23 10892729963841 II25431
8874569 0036159 8910728 31'
24 1099010 9963677 I135333
8864667 0036323 890099036
25
1108726 9963513 I145213
8854787 0036487 88,1274 35
26 1118420 9963348 I155072
8844928 0036652 8881580 34
27 I128092
9963183
1164909
8835091
0036817 8871908 33
28 I137742 9963018 1174724
8825276 0036982 8862258 | 32
29 II47370 9962852
1184518 8815482
9037148 8852630 31
30
1156977
9962686
1194291
0037314 8843023 30
'1166562 9962519 I 204043
8795957
C037481
883343829
32
1176125 9962352 I213773
8786227 0037648 8823875 28
33 1185667. 9962135 12234.82 8776518 0037815
881433327
34
9902017 1233171 8766829 0037983 880481226
12046881 9901849 1242839 8757161 0038151 879531225
1214167
9961681 1252486
8747514 0038319 878583324
37
1262112
12236241 9961512
8737868 0038488 8776376 23
1233061 9961343 1271718 8:28282 0038657 876693922
39 1242477 9,961174 1281 303 8718697 0038826 87575?321
40 1251872 746 1004 1290868
8709132 0038996 8748128 , 20
1261246 9310134 3C0113 8699587 0039166 8738754 19
42 1270600 9960063 1309937
86усов 3 0032337 8729400 I
4,3 1279234 9960-4) 21319442 8680558 0039508 87.0066 1.7
44 1289247 9950;21 1328926
8671074 0039675
8710753 16
45 12985,39 9960149 1338391 8061609 0039857 876146I 15
46 1307812 9959977 1347835 8652165 CO., 0023
8602188 1
47 1317064 9959804 1357260
CC40196 3682936 13
48 1326297 | 9959631 1366665
86333.35
0010369 86737c3 12
49 1335509 9959453 1316051
0040142
8064401 II
50 13-4702 9954284
1385417 8614583
2040716 805523 10
SI 1353875
9959111 1394761
E605236
cc4c889
Com«,125
52
1363028 9758936 1404092 8595908 CoSICH
8636672
53 13;2161
9950701
I413400 8;16600
C041239
7
1381275
S-
9958560
1422089
577,31I
OCTAT
Ó
3012725
55 1390370
143199 del8o41 CO?! 9
8609630 5
56
1399443
7958235 1441210
8556790 Cu41765 Eoço5551
1408501 9958359
1450442 859558 C.CHIVAS
Y
aj 03499
141 7537
9957882
1454055
8540345
C04?118 8502463
59 1426555 9.257705 1408849 8131191 C%20:5
85734-5
60
1435553
1470025 8521975 CC2472 85644471
B
いい​NNN
8805709
31
1195.188
I
35
36
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38
41
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8627839
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57
58
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9957528
26
A Table of Artificial Sines,
8 Degrees.
81 Degrees.
Tang.
Secant.
1
M
Sine.
I
2
18 ) ว
9956993
9956456
IO
9955188
13
I4
15
9954822
18
9954087
20
1657737
21
22
23
30
31
019.1435553 9.9957528 9.1478025 10.8521975 10.0042472 ; 10.8564447 60
1444532
9957350
1487182 8512818 0042650 855546859
1453493
9957172
149632 1 8503679 0042828 8546507 58
3
1462435
1505441
8494559
0043007 8537565 57
4
1471358
9956815
IS14543 8485457 0043185 852864256
5 1480262 9956635 1523627 8476373 0043365 8519738 55
6 E489148
1532692 8467 308 0043544 8510852 54
7 1498015 9956276
1541739 8458261 0043724 8501985 53
8 1506864
9956095 1550769
8449231 0043905 849313652
9 1515694 9955915 1559780
8440220 0044085 848430651
1524507 9955734 1568773
8431227
0044266 847549350
11 1533301 9955552 1577748
8422252
0044448 8466699 | 42
I 2
1542076
9955370 1586706
8413294
0044630 8457924 48
1550834
1595646
8404354 0044812 8449166 | 47
1559574 9955005
1604569 8395431
0044995 8440426 46
1568296
IÓ13473 8386527
0045178
8431704 | 45
16
1577000
9954639
1622361 8377639 0045361 8423000 | 44
17 1585686 9954455 1631231 8368769 0045545 848431443
1594354 9954271
1640083 8359927
0045729 840564642
19 1603005
1648919
8351081
0045913 8396995 41
1611639 9953902
8342263. 0046098 8388361 40
1620254
9953717
1666538
8333462 0046283 837974639
1628853
9953531
167.5322
8324678 0046469 8371147 | 38
1637434
9953345 1684089 8315911 0046655 8362566 37
24
1645998 9953159 1692839 8307161 0046841 8354002 36
25
1654544
9952972 1701572 8298428 0047028 834545635
26
1663074
9952785
1710289 8289711 0047215 8336926 34
27
1671536 9952597
1718989 8281011 0047403 8328414 33
28 1680081
9952409 1727672 8272328 0047591 8319919 32
29
1688559
9952221 1736338 8263662 0047779 8311441 | 31
1697021 9952033 1744988
8255012 0047967 8302979 30
1705465
175 3:622
8246378 0048150 8294535 29
32
1713893
1762239 8237761 0048346 8286107 28
33 1722305
9951464
1770840 8229160 0048536 8277695 27
34 1730699
9951274
1779425
0048726 8269301 26
35 1739077
1787993
0048916 8260923 | 25
17474.39
9950893 1796546
0049107
8252561 24
1755784
9950702
1805082 8194918 0049298 8244216 23
2764112
9950510
1813602 8186398 0049490 8235888 22
39 177 2425
1822106 8177894 0049682 8227575 21
40
1780721 9950126 1830595 8169405 0049874 821.9279 20
41 1789001 9949933
1839068 8160932 0050067 8210999 19
42 1797265 9949740 1847525
0050260 8202735 18
43
180551-2
1855966
0050454 8194488 17
44 1813744 9949352
8135608 0050648 8186256 16
45 1821960
9949158
1872802 8127198 0050842 8178040 15
46
1830160
1881196 8118%24 0051036 816984014
47
1838344
9948769
1889575
8110425 0051231 816165613
48 1846512 9948573 1897939 8102061
0051427 855348812
49
1854665 9948377
1906287
8093713
005 r623 8145335 II
50
1862802
1914621
0051819 813719810
1870923
9947985
1922939
8077061 0052015 8129077
9
1879029
1931241
8068759
005221.2 8120971 8
53
1887120
9947591 1939529
8060471 0052409
8112880 7
54
1895195
9947393
8052198 0052607 8104805 6
55 1903254 9947195 1956059
8043941
0052805
8096746 5
1911299
9946997 1964302 8035698 0053003
8088701
1919328 9946798 1972530
8027470
0053202
8080672 3
1927342 9946599 1980743
0053401 8072658
59 193,5341 9946399 1988941
0053601
8064659
бо 1943324 9946199 1997125
80028751
0053801 805667610
9951844
9951654
a w co w w ew to w to
9951084
8220575
8212007
8203454
36
37
38
995031.8
8152475
8144034
9949546
1864392
9948964
99481&I
8085379
51
52
9.947788
1947802
aco
Mongoleil
4
56
57
58
NOO
8019252
8011059
2
I
Tangents, and Secants.
27
✓
9 Degrees.
୨
M
Sine.
80 Degrees.
Secant.
Tang.
1
1899
I
4
an
II
0056434
13
14
15
2061309
17
T
09.1943324 9 9946199 9.1997125, 10.8002875 10.0053801 10.805667660
1951293 9945999 2005294 7994706 OOS4001 8048707 59
2. 1959247 9945798 2013449 7986551 0054202 8040753 58
3 1967186 9945597 2021588 7978411 0054403 803281457
4 1975110 9945396 2029714 7970286 0054504 802489056
5 1983019 9945194
2037825 7962175
0054806 801698155
6
1990913 9944992
2045922 7954078 0055008 8009087 54
7 1998793
9944789 2054004 7945996 0055211 800120753
8 2006658 9944587 2062072 7937928 0055413
799334252
9
2014509
9944383 2070126 7929874 0055617 79854915E
TO 2022345
9944180
2078165 7921835
0055820 7977655 50
2030167 9943975 2086191 7913809 0056025 7969833 49
I2 2037974 9943771 2094203 7905797 0056229 7962026 48
2045766.
9943566 2102200 7897800
7954231 | 47
2053545 9943361 2110184 7889816 0056639
7946455 | 46
9943156
2118153 7881347 0056844
7938691 45
16 2069059 9942950
2126109 7873891 0057050
793094144
2076795 9942743 213405 I 7865949 0057257
792320543
18 2084516 9942537 2141480 7858020 0057463 7913484 | 42
19 2092224 9942330 2149894 7850106 0057670 790777641
20 2099917 9942122 2157795 7842205
005 7878 79000831 40
21 2107597 9941914
2165683 7834317
0058086 789240339
22 2115263
9941706 2173556 7826444 0058294 7884737 38
23
2122914
9941498
2181417 7818583 0058502 7877086 37
24
2130552 9941289
2189264 7810736 0058711
7869448 36
25 2138176 9941079 2197097 7802903 0058921 7861824 | 35
26 2145787 9940870 2204917 7795083 0059130 7854213 / 34
27 2153384 9940659 2212724 7787276 0059341 7846616 33
28 2160967 9940449 2220518
7779482 0059551 7839033 32
29 2168536 9940238 2228298
7771702
0059762 7831464 31
30 2176092 9940027 2236065 7763935 0059973 7823908 30
31 2183635 9939815 2243819 7756181
0060185 7816365 | 29
32 2191164 9939603 2251561
7748439 0060397
780883628
33 2198680 9939.391 2259289 7740711
ooвoбo9
780132027
34
2206182
9939178
2267004
7732996 0060822
35
2213671 9938965 2274706 7725294
ообщо 35 7786329 | 25
2221147
9938752
2282395
7717605
0001218
777885324
37 2228609
9938538 2290071 7709929
0001462
777139123
38
2236059
9938324 2297735 7702265 0061676 776394122
39 2243495
9930109 2305386 7694614 0061891
77;650521
40 2250918 9937894 2313024 7686976 0062106 7749082 20
41 2258328 9937679 2320650 7679350 0062321
774507219
42
2265725
9937463 2328262 7671738
0062537
7734275
18
43 2273110 9937247 2335863
7664137
772689017
44
2280481
99370.30 2343451 7656549 0062970
45
2287839
9936813 235 1026
7648974 C063187
771216115
46 2295185
9936596
2358582 7641411 0063404 270481514
47
2302518
9936378
2366139 7633861 0063622 769748213
48 2309838
9936160
2373678
7626322 0063840 709016212
49 2317145 9935942 2381203
7618797
0064058 7682855 11
II
50 232444.0 9935723 2388717 761128 0064277 767556010
SI 2331722 9935504 2396218 7603782 0064496 7668278
9
52 2338992 9935285
2403708 7596293
0064715
7661008
8
53
2346249
9935065
2411185 7588815
0064935 7653751 7
54 2353494
9934844 2418650 7581350 0065156
7646506 6
55 2360726
9934624 2426103 7573897 0005376 7639274 5
56 2367946 99.34403 2433543 7566457 0065597 7632054 4
57
2375153
9934181
2440972 7559028
0065819 7624847 3
58
2382349 9933959 2448389
7551611
0066041 761765 1
59
2389532
9933737 2455794 7544206
0066263
7610468
60 2396702 9933515 2463188 7536812 0066485 7603298/ a
7793818 26
به س هی سی با
36
0062753
7719519 / 16
2
2
I
A Table of Artificial Sines,
I0 Degrees.
M
79 Degrees.
2
758899358
7574736 56
9932396
9931946
7539305 SI
9931268
7518189 48
7511173 | 47
7504170 46
1
16
2615779
2630053
7434767 36
20
No
9927362
28
Sine.
Tang.
Secant.
09.2396;02 ,9.7933575 9.2463138 10.7536812 10 0066485 / 10.7603298, le
:
I 2403861
99332)2 2470569
7529431
oc66708
759613959
2411007 9933068 2477939 7522061
0066932
3 2418141
9932845
2485297 7514703
0067155
7581859 57
4 2425264
9932621
2492643 7507357 0067379
5 2432374
2499978 7500022 0067604
7567626 55
6
2439472 9932171 2507301 7492699
0067829
7560528 54
7 2440558
2514612
2485382
0068054
8
755344253
2453632
9931720 2521912
7478098
0068280
7540368 52
9 2460595 9931494 2529200 7470800 0068506
IC 2467746
2536472 7463523
0068732
753225450
II
2474784
9931041
*2543743
7456257 0068959
752521642
ID 2481811 9930814
2550997 7449003 0069186
I3
2488827
9930587 2558240 7441760
0069413
14
2495830
9930359
2565472
7434528 0069641
IS 2502822 9930131 2572692 7427308
006,869
7497178 45
2509803
9929902 2579901 7420099 0070098
17 2516772
7490197 44
9929673 2587099 7412901 0070327
7483228 43
I8
2523729 9929444 2594285 7405715 0070556
2539675
.747627142
19
9929214 2601461 7398539 0070786 746932541
20 2537609
9928984
2608625 7391375 0071016 7462391 49
21 2544532 9928753
7384221 0071247 7455468 39
22 2551444 9928522
2622921 7377079 0071478
7440556 38
23 25563.44 9928291
7369947 0071709
7441656/ 37
24 2565233
9928059
2637173
7362827 0071941
75 2572110 9927827
2644283 7355717 9072173 7427890 35
2578977 9927595
2651382 7348618
0072-405 742102334
27 2585832
2658470 7341530 0072638 741416833
2592676 99;129
2665547 7334453 0072871
7407324 32
2.9 2599509
9.60895
2072613 7327387 007310S
30
7400491 31.
2606330
9920661
2679669 7320331 0273339
739307030
31
2613141
9926427
2636714
7313286
0073573 7386859 29
32
2619941
9926192
219.3749 7300251
0073803
738005923
33
2626729 9925957 2700772 7299228 0074043
7373271 27
34
2633507
9925722 2707786
7292214 0074278
35
2640274
7714788 7285212
0074514 7359726 25
36
2647030
9925250 2721780
7278220
0074750 7.35297024
37 2653775 9925013 2728762
7271238 0074987 734622523
38
2060599
9924776 2735 733 7264267
0075224 7339491 22
39
2667232
9924539 2742694 325 7303
0075461
733276821
40
992.4301 2749644 7250356 0075699 7326055 20
2680647
2756584
7243416
0075937 7319353 12
2687338
9923824
2703514
7236485
0076176 731266213
43 2694019
27704.34 7229566 0076415. 730598117
44 2700689
2777343 7222657 0076654
4.5 2707348
2784242 7215758
0076894 7292652 15
#6
2713997
9972866
2791131 7208869 0077134 7286003 / 14
47 2920035
9922020 2798009 7201991 0077374 727936513
48
2727263
2804878
7195122 0077015 7272737
I 2.
49 2733880 9922144 2811736 7188264 0077856 7266120111
50
2740437
9921902 2818585 7101415
0078098 7259513
51
2747083
9921660
2825423 7174577 0078340 7252917
52 7753609
2.832251 7167749
0078582 7246331
53
9921175 2839070 7760930 0078825 7239755 7
54 2766811 9920932 2845878 7154122 0079068 7233189
55 2773366 9920089 2852677 7147323 0079311 722663; 5
277991 9920445 2859466 7140534 0079555 7220089 4
52
2786445 992020) 2866245 7133755 0079799 7213553 3
58
2792920 9919956
7126986 0080044 7207030
59
2799484 9919711
7120227! 0080289 7200516
be 2805988
99.19466 2886523
7113477 C080534 7194012
7366493 26
9925486
2673945
41
42
9224063
9923535
9923,346
2923106
7299311 16
9922385
II
IO
9921418
2,700-45
2
so
Nai
2
2873014
2879773
I
t
Tangents, and Secants.
29
Į i Degrees.
78 Degrees.
Secant.
M
Sine.
Tang
2812483
2
57
4
Ovog
10
0003009
0083259
0083506
I A
46
17
7072315 41
20
21
7053420 30
23
225
Yo
27
0087304
92805988 9.9919466 9:2886523 | 10.7113477 · 10.0030534 10.7194012
60
|
I
9919220 2893263 7106737 C080780
7187517 59
2818967 9918974 2899923 7100007
0081026
7181033 58
3 2825441
9918727
2906713
7093287
0081273
7174559
2831905 9918480 2913424
7036576 0001520
716809556
5 2838352 9918233
2020126
7072874 0081767
7161641 , 55
6
28.44803
9917986 2926817 7073103 0082014
7155197 | 54
7 2851237
9917737 2933500 7066500 0082263
7148763 / 53
8
2857661 9917489 2940172 7059828
0022511
-7142339 | 52
9 2864076 9917240 2946836 7053164 0082760
713542451
2870480
9916991
2953489 7040511
7129520 jo
II 2876875
9916741 2960134 7039666
7123125 | 49
I2 2883260
9916492
2966769 703.;231
7110740 | 48
13 2889636
9916241
2973395
7026605
0083759
7110364 | 47
2896001 9915990 2980011 7019989 0084010 7103999
15
2902357 9915739 2986618 7013382 0084261
7097643 | 45
16
2908704
9915408
2993216
7006784 0084512
7091296 4+
2915040
9915236
2999804 7000196 0084764
7084900 43
18
2421367 9914984
3006383 6993617 0085016
707803342
19 2927685 9914731 3012954 6987046 0085269
2933993
9914478
3019514 6980486 0085522
7066007 40
2940291 | 9914225 3026066 6973934 0085775
7059709 39
22 2946580
9913971 3032609
6967391 0086029
2952859 9913717 3034143
6960057
0086283
7047141 | 31
24
2959129
9913462
3045667
6954333 0086538
904087136
2965390
9913207 3052183 6947817
0086793
703461035
26
29716 9912952 3058689 6941311
0087048 7028359
34
2977883 9912696 3065187 6934813
7022117
33
28
2984116
9912440 30716,5 6928325 0087560
7015884 / 32
४
29 2990339 2912184 3078155 6921845 0037816
7009661 | 31
30 2996553 9911927 3084626 6915374 0088073 7003447
zo
3002758 9911670 3091088 6908912 0088330
6997242 29
32
3008953 9911412 309 7541
6902459
0088588 Ty910.37
28
33 3015140 9911154 3103985 6896015 ००४४४46
698480027
34 3021317 9910896 3110421
oory104 6978683
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35 3027485 9410637
3116848
0089363 6972515 125
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3033644 9910378 3123266
Jo8y022 6966350 24
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3039794 9910119 3129675
9089881 696020623
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3045934 9909859 3136076
0090141
39 3052066 9909598 3142468
boy0402
by+7934
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0090662
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5935697 19
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3161592 6838408 oc91185 6)29593
I
43. 3070503 9908553 3167950 6832050 0091447 6923497
44 3082590 9908291 3174299
6825701 0091709
6917410 16
45 3088668 9908029 3180640 6819360 co91971 6911332 15
46
3074737 9907766
3186972 6813028 0092234
690520314
47 3100778
9907502 3193295
6800705
oo92498 6899202 13
43
3100849
9907239 3199611 6800389
0092761
6893151
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3112842 9906974 3205918 6794082 0093026 6887108 11
50
3118926 9906710
6787786 0093290
6881074 110
3124951 9906445 3218506
6781494 0093555
6875049
9
52 3130908 9906180
3224788
6775212
0093820
6864032 8
1 53 3136976
9905914 3231001
6768939
0094086
7
$
3142275
9905642
3237327
0094352
55 3148965
9905382
3243584 6756416 0094618
- 6851035
3154947
9905115
3249832 6750168 0094885 6845053
4
57 3160921
3250073 6743927 0095152
3
58
3166885 9904580 3262305 6737695 0095420
59 3172841
9904312 3268529 6731471
0095688
60
9904C44 3274745
6725255
0095956
6821211
31
6889579
6383152
6876734
6870325
6863924
6857532
6851149
6844774
0954066 22
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3128789
30
A Table of Artificial Sines,
77 Degrees.
I
680934158
20 NO
56
6663537
6657409
6614733
21
22
6572434
12 Degrees.
M
Sine.
Tang-
Secant.
09.3178789 | 9.9904044 9.3274745 | 10.6725255 10.0095956 : 10.6821211 60
3184728
9903775
3280953 6719047
0096225 681527259
3190659
9903506
3287153 6712847 0096494
3 3196581
9903237 3293345
6706655 0096763
6803419 57
4 3202495
9902967 3299528
6700472 0097033
6797505 50
5 3208400 9902697 3305704 6694296 0097303 6791600 55
6 3214297
9902426 3311872 6688128
0097574 6785703 54
7
3220186
9902155 3318031
6681969 0097845 6779814 53
8 3226066 9901883 3324183 6675817 0098117 677393452
9
3231938 9901612 3330327
6669673
0098388 6768062 SI
lo
3237802
9901339 3336463
0098661 6762198 50
I1 3243657 9901067 3342591
0098933 6756343 49
12
3249505 9900794 3348711 6651289 0099206
6750495 48
13 3255344 9900521 3354823
6645177 0099479 6744656 47
14 3261174 9900247 3360927 6639073
0099753
6738826 46
15 3266999 9899973 3367024 6632976 O100027
6733003 45
16 3272811 9899698 3373113 6626887 0100302 6727189 | 44
17
3278617 9899423 3379194 6620806
O100577 6721383 43
18 3284416 9899148 3385267
0100852 6715584 42
19 3290206 9898873 3391333 6608667 OIOI127 6709794 / 41
20 3295988
9898597 3397391 6602609 OIO1403 670401240
3301761
9898320
3403441 6596559 O101680 6698239 39
3307527 9898043 3409484 6590516 OI01957
6692473 38
23 3313285 9897766 3415519 6584481 0102234 6686715 37
24 3319035 9897489 3421546
6578454 0102511
6680965 36
25 3324777 9897211 3427566
0102789 6675223 35
26 3330511 9896932
3433578
6566422 0103068 6669489 34
27 3336237 9896654 3439583
6560417
0103346 6663763 33
28 3341955 9896374
3445580
6554420 0103626 6658045 132
3347665
9896095 3451570. 6548430 0103905
665233531
30 3353368
9895815 3457552 6542448 0104185 664663230
31 3359062 9895535 3463527 6536473 0104465 6640938 29
32 3364749 9895254 3469494 6530506 0104746
6635251 28
33 3370428 9894973 3475454 6524546 OI05027 662957227
34 3376099 9894692 3481407 6518593
or05308
6623901 26
35 3381762 9894410 3487352 6512648 O105590 661823825
36 3387418
9894128 3493290
6506710 0105872 661258224
37 3393065 9893845 3499220 6500780 OI06155 6606935 23
38 3398706 9893562 3505143 6494857 0106438 6601294 22
39 3404338
9893279 3511059 6488941 0106721
6595662 21
40 3401263 9892995 3516968 6483032 O107005
6590037 20
41 3415580 9892711 3522869
6477131 0107289 6584420 19
42
3421190 9892427 3528763
6471237
0107573
6578810 18
43 3426792 9892142 3534650 6465350 0107858 657320817
44
3432386 9891856 3540530 6459470 0108144
6567614 16
45 3437973 9891571
3546402 6453598 0108429
6562027 15
46 3443552 9891285 3552267
6447733
0108715
6556448 14
47 3449124 9890998 3558126
6441874 O109002 -6550876 13
48 3454688
9890711 3563977 6436023 0109289 6545312 12
49
3460245 : 9890424 3569821
6430179
0109576 6539755 II
50
3465794 9890137 3575658 6424342 0109863 6534206 10
SI 3471336 9839849 3581487 6413513 OIIOIGI 6528664 9
52 3476870 9889566
3587310 6412690 OI10440 6523130
53
3482397 9889271
6406874 OI10729 6517603
54
3487917 9888982 3598935 6401065 OIIIOI8 6512083
6
55 3493429 9838693 3604736 6395264 OII1307 65065715
56
3498934 9838403 3610531
6389469 O111597 65010664
57 3504432 9888113 3616319 6383681 0111887 64955683
3509922 9887822 3622100 6377900 ΟΙΙ2178
6490078
59 3515405
9887531 3627874
6372126 OI12469 6484595
60 3520880 9887239 3633641 6366352 0112761 6479120
29
1
3593126
7
58
2
1
0
Tangents, and Secants.
31
13 Degrees.
M Sine.
76 Degrees.
Secant.
Tang:
1
2
4
CON and S N
II
13
14
6286333
AAA 1 1 81
17
6263709
0118372
6252437
21
23
9880128
6218775
09.3520880 9.9887239 9.3633641 | 10.6366359 10.0112761 10.6479120 60
3526349 9886947; 3639401
6360599 OI13053
647365159
3531810
9886655 3645155 6354845 OI13345
646819058
3
3537264
9886363 3650901 6349099 0113637 6462736 57
3542710 9886070 3656641 63433.59 0113930
6457290 56
5
3548150 9885776 3662374 6337626 O114224
6451850 SS
6
3553582 9885482 3668100 6331900
O114518
6446418 54
7 3559007 9885188
3673819 63261&I O114812 6440993 53
8 3564426 9884894 3679532 6310468 OI15106 6435574 52
9
3569836
9884599 3685238 6314762 0115401 6430164 SI
IO
35 752470 9884303 3690937 6309063 0115697 642476050
3580637 9884008 3696629
6303371
O115992 6419363 49
12 3586027 9883712 3702315
6297685 0116288 6413973 48
3591409 9883415 3707994 6292006 0116585 640859147
3596785 9883118 3713667
0116882
6403215 46
IS 3602154
9882821 3719333
6280667 ΟΙΙ1179
6397846 | 45
16
3607515 9882523 3724992
6275008 OI17477 6392485 44
3612870 9882225 3730645
6269355 0117775
638713043
18 361.8217 9881927 3736291
0118073 6381783 42
19 3623558 - 9881628
3741930
6258070
637644241
20 3628892 9881329 3747563
0118671 6371108 40
3634219 9881029 3753190
6246810 0118971 6365781 39
22 3639539 9880729
3758810
6241190 OI19271 636046138
3644852 9880429 3764423 6235577 0119571 635514837
24
3650158
3770030
6229970 0119872 6349842 36
25
3655458 9879827 3775631
6224369 0120173
63 4454235
26 3660750 9879525 3781225
O120475
6339250 | 34
27 3666036 9879223 3786813
6213187 OI20777 633396433
28
3671315 9878921 3792394 6207606 0121072
632868537
29 3676587 9878618 3797969 6202031
OI21382 6323413 31
30 3681853 9878315 3803537 6196463 0121685 631814730
31 3687111 9878012 3809100
6190900
0121988 631288929
32 3692363 9877708 3814655
6185345
0122292
6307637 28
3697608
9877404 3820205
OI 22596 6302392 | 27
3702847
9877099
3825748
0122901
6297153 26
35
3708079 9876794 3831285
6168715
0123206 629192125
36 3713304 9876488 3836816
6163184 0123512 628669624
37 3718523 9876183 3842340 6157660 0123817 6281477 23
38 3723735 9875876 3847858 6152142 0124124 6276265 22
32 3728940 9875570 3853370 6146630 0124430
627106021
40
3734139 9875263 3858876
6141124 0124737
6265861 20
41 3739331 9874955 3864376 6135624 0125045 6260669 19
42 3744517 9874648 3869869
6130131
0125352
6255483 | 18
3749696 9874339
3875356
6124644
0125661 625030417
3754868
9874031 3880837 (119163 0125969
3760034 9873722 3886312 6113688 0126278 6239966 15
46
3765194 9873413
3891781
6108219 0126587 623480614
47 3770347 9873103 3897244 6102756 0126897 6229653 13
48 3775493 9872793 3902700
6097300 0127207 6224507
49 3780633 9872482 3908151 6091849
0127518 6219367 11
50 3785767 9872171 3913595
6086405 0127829 6214233 10
SI 3790894 9871860 3919034 6080966 0128140 6209100
52 3796015
9871549
3924466
6075534
0123451 6203985
53 3801129 9871236 3929893 6070107 0128764 6198871 7
54
3806237 9870924 3935313
6064687
OI 29070 6193763
6
55
3811339
9870611
3940727 6059273 0129389 6188661 5
56
3856434 9870298 3946136 6053864 0129702 6183566 4
$ 7
3821523 9869984 3951538
6048462 0130016
58 3826605 9869670 3956935 6043005 0130330
3831632 9869356 3962326 6037674 0130644 6168318
60 3836752 9869041 3967711 6032289 0130959 6163243
33
34
6179795
6174252
43
44
45
6245132 16
5
12
6178477
63.73395
2
59
1
A A Table of Artificial Sines,
14 Degrees.
M
Sine
75 Degrees.
Tang-
Secant.
w
I
602152?
2
6000104
more
9865553
10
II
I 2
13
ar f W
76
5931081
20
21
22
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3951658
3956581
5894310
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;
3841815 9868726 3973089
6026911 0131274
615218:52
3846873 9868410 3978463
OIMI590 6153127150
3
3851924 9868094 3983830 161 70
0131906
614807657
4
38569612 9867778 3989191
CO10839
0132222 6143031 ! 56
S
3862008 9867461 3994547 6005 t 0732539 6.537992 55
6 3867040 9867144 3999896
C132656 613296054
7
3872067 9866827 40052440 5994760
0133173
6127033 133
8 3877087 9866509 4010578 5989422 0131421 6122913 152
9
3882101 9866191 4015910 5984090 0133809 6117899 SI
3887109 9865872
4021237 5978763 0134128
6112891 159
3892111
4026558 5973442
0134447
6107879:49
3897106
9865233
4031873 5963137 0134767 6102894:48
3920096
9864913
4037182 5902818
0135087 6097904 +7
I4 3907079
9864593
4042486
5957514 0135407 6092921 , 46
I5 3212057 9864273
4047784 5952216
0135727
6087943 : 45
I 3917028
9863952
4053076 5946924 0136048 1082972 , 44
17 3921993 9863630 4058363 5941637 0136370 6078007 : 43
18 3926952 9863308 4063644 5936356
0136692 6073048 42
19 39,1905 9862986 4068919
0137014 606809.5 41
3936852
9862663 4074189
5025811 0137337 600314840
3941794
9862340 4079453
5920547 0137660
605820639
3946729 9862017 4084712
5915288
O137983 6053271 38
23
9861993 4089965 5910035
0138307 6C48342 / 37
24
9861369 4095212
5904788
0138631 6043419 / 36
25
3961499
9861045 4100454
5899546 01389.55 0038501 35
26 3966410
9860720
4105690
0139280 603359034
3971315
27
9860394
5889079
4110921
0139606 6028685133
28 3976215 9860069 4116146 5883854 0139931
60237873%
29 3981109 9859742 4121366 5878634 0140258 6018891 '31
3985996
30
9859416 4126581
5873419
0140584 6014004 | 30
31 3990878 9859089 4131789 5868211 0140911 600912229
3995754 9858762 4136993 5863007 0141238 6004246 : 28
4000625 9858434 4142191
5857809
0141566
5999375 27
9858106 4147383
5852617 0141894
4010348 9857777 4152570
5847430 O142223 5989652 25
4015201
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5842248 0142551 5984799 24.
4020048
9857119
4160928 5837072 0142881
527995223
4024889 9856790 4168099 5831901 0143210
5975111 | 22
9856460
4.029724
39
4173265
5826735 0143540
5970276 121
4034554 9856129 4178425
5821575 0143871 5965446 20
4039378 9855798 4183580 5816420 O144202 5960622 19
40441911
0855467 4183729 5811271 0144533
5955204 18
4049009 9855135
43
419.3874 5806126
0144865
5950991 17
4053816
9854803
44
5800987
4199013
0145197
5946184 IÓ
4058617
9854471
4204146 5795854 0145529 5941383 | 15
4063413
9853138 4209275 5790725 0145862
59361587 1.4
4068293 9853805 4214398 5785602 0146195 $931747 13
4072987 9853471 4219515
5780485 0146529
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4077766 9853138 4224628 5775372 0146862 5922714
4082539 9852803
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5770265 0147197
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4087306 9852468 4234838
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O147.5.12 5912694
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4245026 5754974 0148202
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33
34
35
36
37
38
I
40
41
42
45
46
I
II
47
48
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50
SI
工​o9876
e la G
5744806
1
55
56
57
58
I
Tangents, and Secants.
33
74 Degrees.
15 Degrees...
M
Sine.
Tang:
Secant.
o
1
I
2
3
4
5
6
7
8
9
70
1
Хооло
IC
12
13
14
15
16
17
18
19
20
21
22
23
24
75
26
27
28
29
30
31
32.
33
60
9.4129962 9.9849438) 9.4280525, 10.5719475 10.0150562 , 10.5870038
41346.74 9849099 4285575 5714425 0150901
5865326 59
4139381 9848760 4290621 5709379 0151240
5860619/ 58
4144082
9848420 4295661 5904339
0151580 5855913, 57
4148778 9842081 4300697 5099303 OIS1919
585122256
4153468
9847740 4305727 5694273
0152260 584653255
4158152 9847400 4310753 5689247 0152600 584184854
4162832
9847059 4315773
5684227 0152941
583716853
4167506 9346717
4320784
5079211 0153283 5832494 52
4172174 9846375 4325799
567-4201 0153625
5827826 51
4176837 9846033 4330804 5669196 0153967 582316350
4181495 9845690
4335805
5664195 0154310 5&18505 49
4186148
9045347
4340800 5652200 0154653
581385248
4190795
9845004 43745791 5654209
0154996
5809205 47
4195436 9844660 4.350776 5649224 01553-10
5804564 46
4200073 9844316 4355757 5644243 0155684
579992745
4204704 9843971 4360733 5639267 0456029
5795296 | 44
4209330 9843626 4365704 5634296 0156374
579067043
4213950 9843281 4370670 5624330 0156719
5786050 42
4218566
9842935
4375631 5624369 0157065 5781434 | 41
4223176 9842589 4380587
5619413
0157411 577682440
4227780 9842242 4345538 5614462 OI57758
5772220 39
4232380 9841895 4390485
5609515
0198105
5767620 38
4236974 9841548
4395420
5604574 0158452 5763026 37
4241563 9841200 4400363 5599637 0158800
5758437 / 36
4246147 9840852 4405295 5594705 0159148 5753853 35
4250726 9840503 4410222 5589778 0159497 5749274 / 34
4255299 9840154 4415145 5584855 0159846 5744701 33
4259867 9839805 4420062 5579938
0160.195
5740133 / 32
4264430 9839455
4424975 5575025
0160545
5735570 31
4268988 9839105
4429883 5570117
0160895
573101230
4273541
9838755
4134736
5565214 0161245 5726459 29
4278089 9838404 4439685 5560315 0161596
572191128
4282631 9838052 4444579 5555421 0161948
5717369 27
4287169
9837701
4449468 5550532 0162299
5712831 26
4291701 9837348 4454352 5545648 0162652 5708299 25
4296228 9836996 4459232 5540768
0163004
570377224
4300750
9836643 4464107 5535893
0163357 509925023
4305267 9836290 4468978 5531022 0163710 5694733 | 22
4309779 9835936 4473842 5526157 0164064 569022121
4314286 9835582 4478704 5521296 0164413 5635714 20
4318788
9835227 4483561 5516439 0164773 5681212 19
4323285 9834872 4488413 5511587
0165128 5676715 18
4327777 9834517 4493200 5506740 0165483 5672223 | 17
4332264 9834161 4498102 5501898
0165839
5667736 | 16
4336746 9833805 4502940 5497060
0166195 5663254 15
4341223 9833449 4507774 5492226
0166551 5658777 | 14
4345694
9833092
5487398 0166908 5654306 13
4350161
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4354623
4522246 5477754 0167623 5645377 II
4359080
4527061
5472939 0167981 5640920 10
4363532 9831661 4531872 5468128
0168339
5636468
4367380 9831302 45366;8 5463322 0168698 5632020 8
4372 122
9830942 4541479 5458521 0169058 5627578
7
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5453724 0169417 5623141 6
4381292 9830223 4551069 5448931 0169777
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S
4385719 9829862 4555857 5444143 0170138 5614281
4
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5609858
4394560 9829140 4565420
5434580
01 70860 5605440
4398973 9828778
4570194 5429806 0171222 5601027
4403381 9828416 4574904 5425036 0171584 5596619
C
34
35
36
37
38
39
40
41
42
A3
44
45
46
47
4512602
9832377
9832019
48
49
50
51
52
53
54
55
56
57
58
59
nan Q
0.0
34
Å Table of. Artificial Sines,
73 Degrees.
I
5587818 58
56
Ιο
4622423
5544096 48
5539750 47
5535409 46
16
11
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20
21
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549225336
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09.4403381 9.982841694574964) 10 5425036 10.0171584 10.5596619 60
4407784 9828054
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5592216 59
4412182 9827691 4584491 5415509 01 72309
3
4416576 9827328
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5583424 57
4 4420965 9826964 4594001 5405999 0173036
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9 4442837 9825540 4617697 5382303 0174860
5557 163 | 51
4447197 9824774
5377577 0175226
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4451553 9824408 4627145 5372855
0175592
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13 4460250 9823674 4636576 5363424 0176326
14
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4490540 9821092 4669448 5330552 0178908
4494849
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5325873
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5505151 | 39
4499153 9820351 4678802 5321198 0179649
23 4503452 9819979 4683473
5316527 01 80021
5496548 37
24 4507747 9819608 4688139
5311861 0180392
25 4512037 9819236 4692801 5307199 0180764
548796335
26
4516322 9818863 4697459
5302541 0181137
5483678 34
27 4520603
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4702112
0181510
28 4524879 9818117
5479397 | 33
4706762
5293238
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32
29 4529151
9817744
4711407 5288593 0182256
5470849 31
4533418 9817370
4716048
5283952 0182630
546658230
31 4537681 9816995 4720685 5279315 0183005 5462319 | 29
32 4541939 9816620
4725318 5274082 0183380
33 4546192 9816245
4729947 5270053
01837.55
545300827
34 4550441 9815870
4734572 5265428
0184130
26
5449559
35
4554686
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544531425
36 4558926
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5441074
37 4563161 9814740
4748421
5251579 0185260
5436839 | 23
38 4567392 9814393
4753029 5246971
0185637
5432608 | 22
39 4571618
9813986
4757633
5242307 0186014
40
4575840 9813608 47622 3.3 5237767
0186392
41 4580058 2813229 4766829 5233171 0136771
541994212
42 4584271 9812850
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5228579 0187150
18
5415724
43 4588480 9812471
4776009
5223991
44 4592684 9812091 4780592 5210408
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45 4596884 9811711 4785172
5214828
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5403116 | 15
46
4601079 98113311 4789748 5210252 0188669
5392921 14
47 4605270 98109.50
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48 4609456 5810569
5394730 13
4798887 5201113 0189431
12
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49 4613638 9810187 4803451 5196549 0109813
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50 4617816; 98c9805 480801 5191989 OI90195
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35
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532415256
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5291369 48
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20
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60
09.4659353 19.9805963 9.4853390 10.5146610 10.0194037 . IO 5340647
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5 4679960 9804027
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6 4684069 9803639
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TI 4704548 9801690 4902858 5097142 0198310
4708631
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4712710 9800908 491.1802 5038198 0199092
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27 4769380
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36 4805385 9791798 5013588
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37 4809366 2721397
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38
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5182685 21
40 4821283 9790142 5031092 4968908 0204808
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42 4829208 9789,386 5039822 4900178 0210614
43 4833105 9788983 5044182 4955818
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44 4837117 9788579 5048538 4951462 0211421
45 4841066 9708175 5052891 4947104 0211825 9158934 15
46
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47 4848951 9787365 So61586 493871. 0212.635
SI51049 13
48 4852889 9786960 5065928 4934072
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4868595
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36
A Table of Artificial Sines,
18 Degrees.
Sine.
M
71 Degrees.
Secant.
Tang.
o
I
.2
0218752
5092408 58
3
4
5
6
7
8
9
HO
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I 2
9779180
14
4822394
15
16
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2.2
23
24
25
26
27
28
29
30
31
32
33
34
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9.4899824 1 9 9782063 9.5117700 10 4882240 10.0217937, 10.5100176 60
4903710
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4915345
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5134927 4865073 0219582 5084655 56
4919216 9780006
5132210
4460790 O219994 5080784 55
4923083
9779593 5143490 4056510 0220407
507691754
4926946
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4930806
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5069194 52
4934661 9778353 5156309 4843691 0221647 5065339 51
4938513 9777938 5160575
4839425
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4942361
9777523 5164838 4835162 0222477 5057639 | 49.
4946205 9777108
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4950046 9776593 5173353 4826647 0223307
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4957716 9775860 5131855
4818145 0224140 5042284 45
4961545 9775444 5136101
0224556 5038455 | 44
4965370
5190347 4609656 0224974 5034630 43
4969192 9774609 5194.583
4805417 0225391 5030808 142
4973010 9774191 5198819 4801181
0225909 5026990 | 41
4976824 9773772 5,203052 47969.48 0226220 5023176 40
4980635 9773354 5207282
4792712
0226646 5019365 | 39
4984442 2772934 5211503 4788492 0227066 5015558 38
4988245 9772515 5215730 4789270 0227485 5011755 37
4992045 9772095 5219950 4780050 0227905 500795536
49958.40
9771674 5224166 4775834 0228326 5004.160 35
4999633 9771253 5228379 4771621 0228747 5000367 34
5003421
9770832 5232584 4767411 0229163 4996579 | 33.
5007206
9770410
5236795 47632001 0229590 4992794 32
5010987 9769988 5240999 4759001 0290012 4989013 31
5014764 9769566
5245199
4754801
0230434 49852,36 30
5018538 9769143
5249395 4750605 0230857 4981462 i 29
5022308 9768720 5253589 4746411 0231280 4977692 28
5026075 9768296 5257779 4742221 0231704 4973925 | 27
5029838 9767872 5261966 4738034
0232128
4970162 26.
5033597 9767447 5266150 4733850 0232553 4906403 25
503.7353
5270331 4729669 0232978 4962647 24
5041105 9706597 5274508
4725492 0233403 4958895 | 23
5044853
9766171
5278682
4721318
0233829 49.55147
5048598 9765745 5282853 4717147 0234255 4951402 21
5052334 9765,313 5287021 4712979 C254682 4947661 20
5056077 9764841 5291186
4708314 0235109 4943923 19
5059811 9764464 5295347 -3.704633 0235536
4940189 | 18
5003542 9764036 5299505 4700495 0235964 49,645317
5067269 9703608 5303661
4696339 C230392
49.32731 16
5070992 9763179
5307813
4692187 02;6821 4929003 15
5074712 9762750
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4638039 0237350 4925288 | 14
5078428
9762321
5310107
4683893 0237679
4921572 13
5082141 9761891 5320250
4679750 0238109 4917859 12
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5089556 9701030 5328526
4671474 0238970 4910444 10
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5093258 9760599
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5104343 9759303 5345040
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5108031
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5115397
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4642607
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S119074 975 75 70 5361505 4638495 0242430 4880926
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30
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67022 جو
37
38
22
39
40
41
42
43
44
45
40
47
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49
50
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52
53
54
55
56
57
58
59
2
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60
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Tangents, and Secants.
37
19 Degrees.
70 Degrees.
Secant.
M
Sine.
Tang:
18
I
2
بر ط
4
0246354
0246792
0248549
17
9743804
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22
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09.5126419 9.9756701 9.5369719, 10.4630281 10.0243299
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3 5137410 9755394 5382017 4617983 0244606
4 5141067
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16 5184682 9749688 5434994 4565006 0250312
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18
5191904
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5199112
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25 5217074
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26 5220656
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27 5224235
9744806
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28 5227811 9744359: 5483452
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29 5231383
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0256087
30 5 234953
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31 5238518 9743018: 5495500 4504500 0256982
32
5242081
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33
5245640 9742122 5503519 4496481 0257878
34 5249196 9741673 5507523 4492477 0258327
35 5252749 9741224 S5I1525 4488475 0258776
36 5256298
9740774 5515524 4484476
C259226
37
5259844 9740324 5519521 4480479 0259676
38 5263387 9739373 5523514 4476486 0260127
39 5200927 9739422 55%7504 4472496 0260578
40 5270463 9738971 5531492 4468508 0261029
41 5273997 9738519: 5535477 4464523 0261481
42 5277526 9738067 5539459 4460541 0261933
5281053
9737615 5543438 4456562 0262385
5284572 9737162 5547415 4452585 0262838
45
5288097 9736709 5551388
0263291
46 5291614
9736255
5555359 4444641
0263745
47
9735801
5559327 4470673 0264199
48 5298638 9735346 5563292 4436708 0264054
49 5302146
9734891
5567255 4432745
0265109
50 5305650
9734435 5571214 4428786
0265565
51 5309ISI
9733980
5575171 4424829 0266020
52 5312649 9733523 5579125
4420875 026677
53
5316143 9733067 5583077 4416923 0260933
54 5319635
9732610
5587025 4412975
0267390
55 5323123
97321521
3590971 4409029 0267848
50 5320608
9731694 5594914 4105086 0268306
57 5330090
9731236
5540354 4401146 0268764
58 5333569 9730777 5602792 43972081 0269223
59 5337044 9730318
5606727
439 3273 0269632
60
5340517 9729858 Sb10659 4389341 0270143
IO 487358160
4869914 59
486625058
486259057
4858933 56
4855279 55
4851629 54
484798353
484434052
484070051
483706450
48.33431 49
4829802 48
4826176147
4822553 46
4818934 45
481531844
481170543
480809642
4804490 45
4800888 40
4797289 39
4793693 38
479010I 37
4786512 36
4782926 1 35
4779344 34
477576533
4772189 32
4768617 31
4765047 30
4761482 29
4757919'28
4754360 27
4750804 26
4747251 25
4743702 24
4740156 23
473661322
4733073
4729537
4720003 19
4722474 18
4718947 17
4715423 16
471120315
4708;86 14
4704872 13
4701362 12
4697854 IL
4694350 lo
4690849
4687351 8
4083857 7
4680365 6
4676877
5
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4669910 3
4666431
4662956
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21
20
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43
44
4448612
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83
A Table of Artificial Sines,
20 Degrees.
Sine
69 Degrees.
Secant.
MI
Tang.
183
I
2
3
4
5
6
8
9
JO
II
I 2
13
14
IS
16
17
18
19
20
4594097 41
21
9.5340517 i 9.9729858 9.5610659 10.4389341 : 10.0270142 | 10.46594
60
5343986 9729398 5614588 4385412 0270602 465601459
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4652548 58
5350915 9728477
5622439
43775611 0271523 4649080 57
$354375 9728916 5626360 4373640 0271984
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5361286 9727092 5634194 4365806 0272908 4638714 54
5364737 9726629 5638107 4361893 0273371 4635263 53
5368184 9726166 5642018 4357982
0273834 463181652
5371629 9725703 5645925 4354075 0274297
4628371 SI
5375070 9725239 5649831 4350169 0274761 462493050
5378508 9724775
5653733
4346267
0275225
4621492 49
5381943 9724310 5657633 4342367 0275690
4614057 48
5385375 972;845 5661530 4338470 0276155 4614625 47
5388804
9723380
5665424
4334576 0276620 461119646
5392230 9722914 5669316 4330684 0277086 460777045
5395653 9722448 5673205 4326795 0277552 4604347 44
5399073 9721981 5677091 4322909 0275019 4600927 43
5402489 9721514 5680975 4319025 0278486
459751142
5405903 9721047 5684856 4315144 0278953
5409314 9720579 5688735 4311265 0279421
4590686 40
5412721 9720110 5692611 4307399 0279890 4587279 39
5416126 9719642 5696484 4303516 0280358
458387438
5419527 9719172 5700355 4299645 0280828 4580473 37
5422926 9718703 5704223 4295777 0281297
4577074 / 36
5426321 9718233
5708088
4291912
0281767
4573679 | 35
5429713 9717762
5711951 4288049 0282238
457028734
5433103 9717291 5715811
4284189 0282709 4566897 33
5436489 9716820 5719669
4280331 0283180 4563511 32
5439873 9716348 5723524
4276476 0283652 4560127 31
5443253
9715876
5727377
4272623 0284124 4556747 30
5446630 9715404 5731227 4268773 0284596
455337029
5450005 9714931 5735074 4264926 0285069 4549995
28
5453376 9714457 5738919 4261081 0285543
4546624 | 27
5456745 9713984 5742761 4257239
0286016
4543255
26
546οΙΙο 9713509
5746601
4253399
0286491 453989025
5463472 9713035 5750438 4249562 0286965 4536528 24
5466832
9712560
5754272 4245728 0287440
4533168 / 23
5470189 9712084 5758104 4241096 0287916 4529311 22
5473.542 9711608 5761934
4238066
0288392 4526458 21
5476893 9711132 5765761 4234239
02868áv
4523107
20
5480240 9710055 5769585 4230415 0289.345 4519760 19
5483585 9710178 5773407 4226593 0289822
451641518
5486927 9709701 5777226
4222774 0290299
4513073 17
5490266
9709223 5781043 4218957 0290777
4509734 / 16
5493602 9703744 5784858 4215142 0291256 450639815
54969,35 2708265 5788669 4211331 0291735 450306514
5500265 9707786
5792479 4207521 0292214 4499735 | 13
5503592
9707306
5796286 4203714 0292694 449640 12
5506916 9706826 5800090 4199910 0293174 449308411
5510237 9706346 5803892 4196108 0293654
4489763 10
5513556 9705865 5807691 4197309 0294.135 4486444 9
5510871 9705383 5811488 4188512 0294617 4483129 8
5520184
9704902 5815282 4184710 0295098 4479816 7
5523494
9704419 5819074 4180926 0295581 4476506 6
5526801
9703937
5822864 4177136 0296063 4473199 5
5530105
970,3454
5826651 4173349 0296546 4469895 4
5533406
9702970 5830435 4169565 0297030
4466594 3
5536704 9702486
5834217 4165783 0297514 4763296
5539999 0702002 5837997 4162003 0297998 4460001
$543292 9701517 5841774 4158226 0296483 44567080
بی بی بی بی سی در
22
23
24
25
26
27
28
29
30
31
32
33
34 1
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
06
N ah A UJ N
IO
+
1
л л л
2
I
Tangents, and Secants.
39
21 Degrees.
Sine.
68 Degrees.
Secant.
M
Tang.
0
o
I
189
4453419 | 59
2
56
3
4
5
6
7
8
9
IO
4437013154
443374153
443047152
II
48
41
5611779
4064577
36
ب ي د د د د د د
9.5543292 | 9.9701517 9.5841774, 10 4158226 10.0298483 ; 10.4456708 60
5546581
9701032
5845549 4154451, 0298968
5549868
9700547 5849321 4150679 0299453 445013258
5553152 9700061
5853091 4146909 0299939 4446848
57
5556433
2699574 5856859 4143141 0300426 4443567
5559711 9699087 5860624 4139376
0300913 4440289
55
5562987 9698600 5864386
4135014 0301400
5566259 9698112 5868147 4131853 0301883
5569529 9697624
5871904
4128096 0302376
5572796
9697136 5875660 4124340 0302364 4427204 51
5576000
9696647 5879413 4120587 0303353 4423940
50
5579321 9696158 $883163 4116837 0303842 4420679
49
5582579
9695668 5886912 4113088
0304332 4417421
5585835 9695177 5890657 4109343 0304823 4414165
47
5589088 9694687 5894401 4105599 0305313 4410912 46
5592338 9694196 5898142 4101858 0305804 4407662
45
5595585
9693704
5901881 4098119 0306296 4404415
44
5598829 9693212 5905617 4094383 0306788 4401171
43
5602071 9692720 5909351 4090549 0307289 4397929
42
5605310 9692227
5913082 4086918 0307773 4394690
5608546 9691734 5916812 4083188 0308266 4391454
40
9691241 5920539 4079461 0308759
4388221
39
5615010 9690746 5924263 4075737 0309254 438-1990
38
5618237 9690252 5927985 4072015 0309748 4381763
37
5621462 9689757
5931705
4068295 0310243 4378538
5624635 9689262
5935423
0310738 4375315
35
5627904
9688766
5939138
4060862 0311 234
4372036
34
5631121 9688270
5942851 4057149 OZI1730 4368879
33
5634335 9687773
5946561
4053439 0312227 4365665
5637546 9587276
5950269 4049731 0312724 4362454
5640754 9686779 5953975 4046025 031322 I
4359246
5643960 9686281 5957679 4042321 0313719 4356040
29
5647163 9685783 5961380 4038620 0314217
4352837 28
5650363 9685284 5965079 4034921 0314716 4349637
27
5653561 96154785 5968776 4031224 0315215 4346439 26
5656756 9684286
5972470 4027530 0315714
5659948 9683786 5976162
4023838 0316214 4340052
24
9683285 5979852
4020149
0316715
5666324
9682784 5983540 4016460 0317216 4333676
22
5669508 9682283 5987225 4012775 0317717
433049221
5672639 9681781 5990908 4009092
0318212 432731120
5675868
9681279
4005412
0318721 4324132
19
5679044 9680777 5998267 4001733 0319223 432095618
5682217 9680274 6001943
3998057 0319726
5635337
9679771
6005617 3994383
0320229
4314613 1 16
5638555 9679267 6009289 3990711 0320733
5691721 9678763 6012958 3987042 0321237 4308279
14
5694833 9678258 6016625 3983375 0321742 4305117
13
5698073 9077753 6020290
3979710 0322247
5701 200 9677247
6023953
3976047 0322753 4298800
II
570-13.55 9676741 6027613 3972387 0323259 4295645
5707506 9676235
6031271 3968729 0323765 4292494
5710656 9675728 6034927 3965073 0324272
4289344
5713.02
9675221 60385 81 3961419 0324779 4286198
7
5716946
9674713 60112233
3957767
0325287 4283054 6
5720087
9674205
6045882
3954118
0325795 4279913
5
5723226 2673697
3950471 0326303 4276774
4
5726362 9673188
6053174
3946826 0326812 4273638
3
5729495 9672679 6056817
3943183 0327321 4270505 20
5732626 9672169 6060457 3939543 0327831
5735754 9671659 16064096
3935904
0328341 4264248
I2
13
14
IS
16
17
18
19
20
21
22
23
24
25
26
27
28
22
30
31
32
33
34
35
36
37
38
39
40
41
42
43
41
4.5
40
47
48
49
50
SI
52
53
54
55
56
57
58
59
32
31
30
434324425
5663137
4336863 23
5994588
200 N
431778317
4311445 15
4301957 12
10
6049529
4267374
I
60
4
A Table of Artificial Sines,
.
22 Degrees.
67 Degrees.
Secants.
Tang.
Sines.
M
O
I
2
3
5
6
8
9
10
I 2
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
O on an f W
Co w w w cs ta ta ta 65 W 19 W D N N N N
Van die w
9.5735754 | 9.2671659 9.6064096 | 10.3935904 10:0328341 10.426424660
5738880 9671148 6067732
3932268
0328852 4261120 52
5 742003 9670637 6071366
3928634
0329363 4257997 | 58
5745123 9670125 6074997
3925003
0329875 425487757
5748240 96696,14 6078627 3921373 0330386 425176056
5751356 9669101 6082254
3917746
0330899 4248644 55
5754468 9668588
6085880
3914120 0331412 4245532 54
5757578 9668075 6089503 3910497 0331925 424242253
5760685 9667562
6093124
3906876 0332438
423931552
5763790 9667048 6096742
3903258 0332952 4236210 51
5766892 9666533 6100359
3899641 0333467 423310850
5769991 9666018
6103973
3896027 0333982 423000949
5773088 9665503 6107586
3892414
0334497 422691248
5776183 9664987 6111196 3888804 0335013
422381747
5779275 9664471 6114804 3885196 0335529 4220725 46
5782364
9663954
6118409 3881591
0336046
4217636 | 45
5785450 9663437 6122013 3877987 0336563
4214550 44
5788535 9662920 6125615 3874385 0337080
4211465 43
5791616 9662402 6129214
3870786 0337598
4208384 | 42
5794695 9661884 6132812 3867188 0338116 42053054I
9661365
6136407
5797772
3863593 0338635
4202228 40
5800845 9660846 6140000 3860000 0339154 419915539
5803917 9660326 61.43591 3956409 0339674 4196083 38
5806986 9659806 6147180 3852820 0340194 4193014 37
5810052 9659285 6150766
3849234 0340715 4189948 36
5813116 9658764 6154351 3845649 0341236
4186884 35
5816177
9658243
6157934
3842066
0341757
4183823 34
5819236 9687721 6161514
3838486 0342279
4180764 33
5822292 9657199
6165093
3834907
0342801
4177708 32
5825345 2656677 6168669
3831331 0343323
4174655 31
5828397
9656153
6172243
3827757
0343847
4171603 30
5831445
9655630 6175815 3824185 0344370
416855529
5834491
9655106
6179385 3820615 0344894 4165509 28
5837535
2654582
6182953
3817047
0345418
4162465 27
5840576 9654057
6186519
3813481 0345943 4159424 26
5843615 9653532 6190083 3809917
0346468 4156385 | 25
5846651 9653006
6193645 3806355 0346994 4153349
24
5849685
9052480
6197205
3802795
0347520 4150315
23
5852776 9651953 6200762 3799238 0348047
22
4147284
5855745 9651426 6204318
21
0348574 4144255
5858771
9650899 6207852
3792128 0349101 4141229
5861795
9650371 6211423 3788577 0349629
413820519
5864816 9649343
3785027 0350157 4135184 18
5867835 9649314
0218520
3781480 0350686
4132165 17
5870851 9648785 6222066 3777934 0351215 412914916
5873865 9648256
6225609 3774391 0351744
412613515
9647726
6229150
5876876
3770850 0352274 4123124 14
5879885 9647195 6232690 3767310 0352805 4120115
13
5832892 9646665 6236227 3763773 0353335 411710812
5835896 9646133 6239763 3760237
4114104
0353867
5888897 9645602 6243296 3756704
4III103
0354398
5891897 9645069 6246827 3753173 0354931 4108103
9
9644537 6250356
58:14893
3749644
4105107
0355463
8
6253884 3746116
5897938 9544004
0355996 4102II2
7
6257409 3742591
0356530
4099120
5900880 9645470
6
6210932
5903569 964.2437
3739068 0357063
4096131
5
6264454
3735546
5906356: 9642402
0357598 4093144 4
3732027 0358132
4090159 3
5909841: 9641868
9641332
5912323
6271491
4087177
3728509
3724994
9640797 6275006
0359203
4084197
3721481
5918780: 96402611 6278519
0359739
4081220
38
3795682
20
6214973
39
40
41
42
43
44
45
46
47
48
49
50
II
IO
51
52
53
54
95
56
57
58
59
oo
6267973
0358658
2
I
5915803
Tangents and Secants.
41
23 Degrees.
66 Degrees.
M
6 2 1 0 8
38
3
4
5
(
ე
8
ვნ)3444
4
9
IO
I
II
6317037
I2
I 3
14
IS
403978446
I
17
18
19
20
21
2 2
23
23
25
26
N 1 / N
Sine.
Tang.
Secant.
9.59187730 9.9640261 9.6278519, 10.3721481 100359737 10.408122060
,
5921755 2639724 6282031
37 (79) o369276 4078245 52
5924728 9639187
62855410
3714460 0360813 4075272
5927698 938650 6289048 3710952 0361350
4072302 57
5930666 9638II2
6292553 3707447 0361883 4069334 / 56
5933631 96375 74 6296057 370394,3 0,2426 4066369 | 55
5936594 937036 6299558 3700442 0362964 406340654
5939555 9636496 6303058 3696942 o3(3504 406244s | 5 3
5942ა
9635957
6306556
0,4043
405 748752
5945464
9635457 6310052 3589918 0364583
425 45 34 | 5I
5940422 9634877 6
313545
3686455 0,652
4051578 | 50
5951373 9634336
3032963 03:55664 48627 | 42
5954322
96 337:25 6320527 3079473
(364)205
404567843
5957268
963.3253 6324ors 3675985 0366747
40427,3247
5960212 96327 II 632750I
3672494
035722)
5963654
96 32 16გ
6330985 3,5751 5 0367832
403634645
5966993 963 1625 633446
2665532 0368375
4033207 44
5969030 9531082 6337948 3602052 cკზ918
403027043
5);1965 950 38 6341426
3658574
0369402 4c28635 | 42
5974897
90 29)24 6344903 365 5097 0370006
4025Icy | -4I
5777027 962 $24) 6348378 ვი; 1622 0370551
40221731 40
598e7c4 9628904 6351955 304815 c3/10) 401924639
598379 9628358 6355321 3644679
c371642 4016;21.38
5986602 9627812 6358790 3641210 0372103 40I 3398 6 37
595)523 2627266 6362257 3637743 0372734
43I_457] 36
5992441 9626719 6365722 3634278 03:23
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5995357 96-6172 63631s 3636815
0,37,3828 40046-43 34
5998270 902 (24 6372646 3627354
0374376 4OIT კბ33
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3623894 0374924 399881932
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3,538012 0388772 3926784
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3534600 0389332 3923932
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3531190 0389892 32-1082
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60017659 9609548: 6472217 3.527783 C39245 24 3918235 4
(084614 9089*7 647562- 352437 O:9Ion 3 3). 3) 3
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35 I:414) e392c58 3კისას): 0
(
28
29
30
SI
32
33
34
35
ვ)
37
38
39
სა და
6410577
4I
)
4 2
}
43
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3 2
I
53
54
5 S
56
57
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42
A Table of Artificial Sines,
24 Degrees.
65 Degrees.
MI
Sine.
Tang.
Secant.
o
180
I
2
3904031 159
58
3
4
5
6
7
8
8895535 ' 56
38.72707 55
3
6516359
9
10
11
12
13
I 4
IS
16
17
28
19
20
21
22
23
24
9527676
3449888
9594821
3844976/ 38
3839401 36
25
26
27
28
29
30
31
32
i Q CS P A
9591380
9.6.09313319.9607302
60,5969 9601739
60-18803 9606176
6101635 9605612
6104465 9605048
6107293 9604484
6ΙΙοΙ18 9603919
6112241 9603354
6115762 9602788
6118580 9602222
6121397 9601655
6124211 9601088
6127023 96005 20
6129833 9599952
613264.1
9.599334
6135440 9598815
61382501 9598246
6141051
61438501 9597106
6146647 9596535
6149441 9595964
6152234
9595393
6155024
6157812 9594248
6160599 9593675
61633821 9593102
6166164
9592528
6168944
9591954
6171721
6174496 9590805
6177270
9590229
6180041 9589053
6182809 9509077
61855701 0588500
61%8341 9587923
6191103 9587,345
619.864 | 9586761
6196622
6199378 9585609
6202132 9545030
6204664 9504450
6207634 958366y
6210382 9583283
6213127 9582707
6215871 9582125
6216612 9581543
6221351
5580961
6224088 9560378
62,26824 9579704
6229557 95 79210
62322871 9578626
6235010
9578041
6237743
9577456
6240460 9576870
6243100
9576284
6245911 9575697
6248629
9575110
6251346
9574522
6254060 9573934
6250772
9573346
62.594831 9572757
60
26485831, 10.3514169 . 10.0392698 10.3906867 ?
6489230 3510770 0393261
6492628 3507372 0393824 3901197
6496023 3503977 0394388 3898365 1 57
6492417 3500583 0324952
6502809 3497191 0395516
)
6506199
3493801 0396081 388y88254
6509587 3490413 0396646 3387059 53
6512974
3487026 0327212 3884238; 52
3483641 0397778
3381420 51
6519742
3480258
0398345
3878603! 50
6523123 3476877 0398912 3875789 49
6526503 3473497 0399480
3872973 48
6529831 3470119 0400043 387016747
6533257 3466743
0400616 3867359
46
6536631 3463369 0401185 3864554 45
6540004 3459996
0401754
3861750 44
6543375 3456625 0402324
3858949 | 43
6546744
3453256
0402894
3856150 42
6550112
0403165
3853353 | 41
6553477 3446523 0404036
385055940
6556841 3443159
0404607
3847766 32
6560204
3439796;
0405179
6563564 3436436 0405 752
3842188/ 37
6566923 3433077!
0406325
6570280 3429720
0406898
383601835
6573636 3426364 0407472
3833836 | 34
6576989 3423011 0408046
3831056 33
6580341 3419659 0408620
3828279 32
6583692 3416308 0409195
3825504 31
6587041 34129591 0409771
3822730 30
6590387 3409613 0410347
391995929
6593733 3406267 0410923 3817191
6597076 3402924 0411500 3 3814424 27
6600413 3399582
0412077
3311659
26
6603758 3390242 0412655
3308897 1 25
6607097 3392903 0413233
3806136 24
66107.34
3389566 0413712 38033781 23
6613763 3381231 0414391
3800622 | 22
6617103
3382897 0414270
6620434
3379506
0415550
6623765 3376235
0416131
3792366 19
6627093
0416712
3372907
18
3789618
6630420
3369580
0417293
3786873
17
3366255
0417875
6637069 3362931 0418457
37813118 | 15
335900y 0419039
377864914
6643711
04.19622
6647030 3352970 0:20206
6650346 3349654 0420790 3770413
6653662 3346338 0421374
3767713 10
6656975 3343025
0421959
37649824
6660288 3339712 0.122544 3762252
8
6663598
3336402
0423130 3759532
7
6560907 33,3309,3 0423716 3750310 6
6670214
3329786
0424303
3754089
5
3326481
042-1890
3751371 4
6676823 3323177 0425478 3748654 3
6660126
3319874
3745940 2.
6683426 3316574
042665-4
3743222
6686725 3313275 04272431 3740517
28
33
II
34
35
& w w w w w w w ca a
361
37
9586188
38
379766821
3795116 20
39
40
41
42
43
6033745
3784129 16
IG
6640391
44
4.5
46
47
48
49
SO
51
33,5628,
3775912 13
3773176 12
II
52
53
54
55
6
57
58
$9
60
a la la la
6673519
0421066
I
Tangents, and Secants.
43
1
25 Degrees.
64 Degrees.
M
Sine.
Tang.
Secant.
O
I
3737809 59
2
3729697/ 56
3718910 52
1
371352350
3
4
5
6
7
8
9
IO
II
I
13
14
IS
16
IT
18
19
20
21
22
3705471 47
370278946
37001IO | 45
3
24
25
26
27
2
28
29
30
31
32
33
34
35
36
37
9.6259483, 9.9572757 9.6686725 F10 3313275 10.0127243 10.3740517
60
6262191
9572168
6690023
3309977 0427832
6264897
9571578
6693319 3306681 0428422 3735103
58
6267601
9570988
6696613 3303387 0429012 3732399
57
6270303
95 70397 6699906 3300094
0429603
6273003
9569806 6703197 3296803 0430194
3726997 55
6275701
95692151 6706486
3293514 0:430785
54
3724299
6278397
9568623
6709774 3290226 0131377
3721603 | 53
6281090
9568030
6713060 3286940 04.;1970
6283782 9567437
6716345 3233655 0432563
3716218 51
6286472 9566814 6719628 3280372
0433156
6289160 9566250
6722910 3277090 0433750
3710840| 49
6291845
9565656
6726190 3273810 0434344
370815548
6294529
9565061
6729468 3270532 0434939
6297211
9564466
6732745 3267255
0435534
6299390 9563870 6736020 3263980 0.136130
6302568 9563274 6739294 3260706 0436726 3697432 44
6305243
9562678 6742566
3257434 0437322
359475743
6307917
9562081
6745836 3254164 0437919
3692083 42
6310589 9561483
6749105 3250895 0.138527
3639411 41
6313258
9560886
6752372
3247628 032514
3686742 | 40
6315926
9560287 6755638 3241.362
0.139713
3081074
39
6318591
9559689
6758903 3241097 0440311 3631403
38
6321255
9559089 676.165 3237835 0440.)II 3678745 | 37
6323916 95584.90 6765426
3234574 0441510
3676084 36
6326576 9557890 6768686
3231314 0142110 3673424 35
6329233
9557289
6771944
3228056 0442711 3670767 / 34
6331889
9556688
6775201 3224797 0443312 3668111 33
6334542
9556087
6778456 3221544 0443913 3665458 32
6337194
9555485
6781709
3218291 0444515 3662806 31
6339844
9554882
6784961 3215039 0445118 3660156 / 30
6342491
9554280
6788211 3211789 0445720 3657509 29
6345137
9553676
6791460 3208540
0446324 3654863
28
6347780
9553073 6794708 32052)2 0110927
3652220 127
6350422
9.552409 6797953
26
3202047 • 04147531 3649578
6353062
6801198 3198802 0448136
36469.38 25
6355699
9551239
6804440
3195560
0448741
3644301' 24
6358335 9550653 6807682 3192318 0742,347
3641665 23
6360969
9550047
6810921 3189079 0449953
6363001
9549441
6814160 3185840 0450559
6366231
9548834 6817396 3182604 0451166 3633769
6303859 9548227
6820632 3179368 0451773
3031141 19
6371484 9547619 6823865 3176135 0452381
6374108
95-7011
6827098 3172902 0452989
3625892 17
6376731 9.546402
6830328
IÓ
3869672 0453598 3623269
6379351
9545793
6833557
3166443 0454.207
3620649 15
6381969
9545184
6836785 3163215 0454815 3618031 14
6384585
954-574
6840011 3159989 0455426 3615415. 13
6397199
9543963
6843236 315676_1 0450037
3612801 12
6389812
9543352
63,6459
3153541 0456648
6392422
9542741 6847601
3150319 0457259
6395030 9542129
6852901 3117099 0457871 3604970
6397637
9.541917
6856120
8
3143880 0458483 3602363
6400241
9.540904 6859338 3140662 0459096 3599759
7
6402844 9540291
6
6802553 3137447 0459709
3597156
6405445
9539677
6865768 3134232
0460323
3594555
5
6408044
9539063
6868981 3131019
0.460937
357 1956
4
6410640 95,38448
6872192
3127808
0461552
589360 3
6413235 95.378,33
3124598 0462167
3586765
2.
6415828
9537218 6878611
3121389 0462782 3504172
6418420
9536602
6881818 3118182 0463398
9551864
38
3639031 27
363639921
20
3628576 18
39
40
41
42
43
44
45
46
4.7
48
49
50
3610188 Ir
360757810
51
52
53
54
55
56
57
58
59
60
6875402
I
3581580
44
A Table of Artificial Sines,
26 Degrees.
63 Degrees.
Sine,
Tang.
Secanit.
M
1
2
6421009
04-3596
6420132
6431347
6-43.3426
6436504
0439080
6441054
2 I
22
9.6418420, 9.9536602 9.6881818 10.3118142 10.0463398. 10.3581380 60
9535985
6085023 3114977 0464015 357899159
9535369
0888227 3111773 0464631 357640458
3
9534751
0891430 3108570 0465242
357381857
4
0428765
9534134
6894031
3105,369
0465867 35712355
5
9533515 6897831
6
3102169 0466489 356865355
9532897
6901030 3098970 0467103 356007454
7
8
9532278
6904226 3045774 0467722 3563496 | 53
9531658
6907422
3092578
0468342 356092052
9
9531038
6710616 3089,384
046sy02
IO
355834651
6444226
9530418
6913809 3086191
II
0409582 3555774 JO
6446796
9527797 6917000 3083000
I2
0470203 355320449
0449365
9529175
13
6920189
3079811
0470825 355063548
6451931
9528553 6923378 3076622
04714 17 354806747
14 6454496
95279.31 6920505 3073435 0472009
15
354550440
6457058
9527308
6929750
16
3070250 0472692 354294245
6459619
952668
6932934
3067066
0473315 354038144
17
6462173
9520001
6936117
18
3063883 0473939 353782243
6464735
9525437 6939208 3000702 0474563 353526542
19
6407290
9524813 6942478
3057522 0475187 353271041
20
6469844
9524188
6945656
3054344 0475812 353015640
6472395
9523562
6448833
3051107 0476438 352760$ 32
6474945
9522936
6952009 3047991 0477064 3525055 38
23 6477492
9522310 6955183 3044817
0477690
3522508 37
24 6480038
9521683
6958355 3041645 0478317 3519962 36
25 6482582
9521055
6961527
3038473 0478945 3517418 35
26
6485124
9520428
6964697 3035303 0479572 351487634
27 6487665
9519799 6967865 3032135 0480201
28
3512335 33
6490203
9519171
6971032
3028968 0480829 3509797 32
29
6492740
9518541 6974198
3025002
0481459 350726031
30
6495274
9517912 6977363 3022637 0482c88 3504726 / 30
31
6497807
9517282
6980526 3019474 0482718 3502193 | 29
32
6500338
95116GI
6983687 3016313 0483349
3499662 / 28
33
6502868 9516030 6986847 3013153 0483980 3497132 27
34
6505395
9515389
6990006 3009994 0484011 349400526
35
6507920
9514757 6993164 3000836 0485243 349208025
36
6510444 9514124 6990320 3003600 0485876 348955624
37
6512966 9513492 6999474 3000526 0400508 348703423
38
6515486 9512858 ४ 7002628
2997372 0487172 3484514 22
39
6518004
9.512224 7005780
2994220 0487776 3481996 | 21
652052T 9511590 7000930 2991070 0485410 3479479 20
0523035
9510956
7012080
2907920 04840914 3476965. 19
6
52554% 9510320 7015227 2987773 0489680
347445219
6527059 9509685 7018374 2981626 0490315 3471941
!
1530568 9500049 7021519 2478481 04909.51 346243210
6533075 9508412 7024063 2975337 0491588
3461,925 19
IS
0535581 9507775 7027805 2972195 04422451
3464419 114
6538084
9507138 7030946 2409054 0092862
3461916 13
6540586 9506500 7034086
2465914 0493500 3459114 12
6543086 9505861 7037225
2962779 0494139 34561114 II
6545584 9505 223 7040362 2959038 0494777 3454416:10
6548081 9504583 7043497 2956503
0425417 34519.19
12
6550575 9503944
7046632
205336€
0496096 3147425
0559068 9503303 7049765 295025 0446677 3446932
6555559
9502663 7052897 29-7103 0497337 3444441 6
6558048
9502022 7036027
24;43273
0497973 3141952 5
6560536
7059756
2040044 C498020 3439.04
4
6563021
9500738
7002284 2937716 0499262 3436979 }
6565505
9500095 7065410 29,34590 0499905 3434495
6507987
9499452 7068535 2031165
I
C00548 3432013
bu 65704160 9-11.0009
7075654
292821
05501IO 3429532
1
در لحد کا ما
30
I
42
I
43
44
4.S
qú
47
8
49
o
5I
52
53
54
55
51
9501380
57
58
و کمر
1
Tangents, and Secants.
45
27 Degrees.
Sine.
62 Degrees.
M
Tang.
Secant.
| 9 ១១
2
O OCard UJ NM O
58
342210257
3419629 56
6
52
SI
IO
II
IZ
$8
!
3397450 | 47
46
45
17
7124562
Ig
867237
20
21
22
2859949
3375414 / 38
23
2844492
3). 6570408 9.9498809 9.7071659,10.2928341 10.0501191, 10.3429532
6572946 1 9498165 7074781 2225219 0501835 3427054
59
6575423
9497521
7077902 2922098
0502479 3424577
3 6577898
9496876 7081022 2918978
OGO3124
4 6580371 9496230 7084141 2915852 0503770
5 6582842
9495585
1087258 2912742 0504415
341715855
6
6585312
9494938
7090374 2909626 0505062
341468854
6587780 9494292
7093483
2906512 0505708 341 2220 53
8
6590246 9493645 7096601 2903392
0506355 3409754
9 6592710 9492997 7099713 2900287 0507003 3407290
6595173 9492349 7102824 2897176. 050765 I
340482750
6597633 9491700 7105933 2894067 0508300
3402367 49
6600093
9491051 7109041
2840959
0508949 3399907
13
6602550
9490402 7112148 2887852 0509598
14 6605005
9489752 7115254
2884746 0510248
3394995
IS 6607459
9489101 7118358
2881642 OS10892 3392541
16
6609911 9488450 7121461
2878539 0511550
3390089 44
6612361 9487799
2875438
0512201 -338763943
18 6614810 -9487147
7127662
2872338 0912853
338519042
6617257
94.86495 7130761
0513505
338274341
6619702
9485842
7133859
2856141
0514158 3380298 40
6622145 9485139 7136956
2863044
0514811 3377855 39
6624586 9484535 7140051
Og15465
6627026
9483881
7143145
2856855 0516119
3372974 37
24 6629464 9483227 7146237 2853763 0516773
3370536 36
25 6631900
9482572 7149329 2850671 0517428 336810035
1 26 6634335
9481916
7152419 2847581 0518084 3365665 / 34
27 6636768
9481260 7155508
0518740
336323233
28 6639199
9480604 7158595 2841405 0519396 3360801 32
29 6641628
947997 7161682
2.838318 0520053 335837231
30 6644056 9479289 7164767
2835233
0520711
3355944 | 30
31
6646482 9478631
7167851
2832149
0521369
335351829
32
6648906
9477973 7170933 2829067 0522027
3351094
28
33 6651329
9477314 7174014 2825986
0522686
334857127
34 6653749
9476655 7177094 2822906 0523345
35 6650108
9475995 7180173 2819827 0524005
334383225
36 6658586
9475335 7183251
0524665
3341414 24
37
66610OI
9474674 7186327 2813673 0525326 3338999 / 23
38
6663415
9474013 7189402 2810598 0525987 3336585 | 22
39 6665328
1473352 7192476
2807524 0526648
3334172
40 6663238
9472689
7195549
2804451 0527311
3331762 20
41
6670647
9472027 7198620 2801380 0527973
42
6673054
3329353 19
7201690 2798310 0528636
43
6675459
9470700 7204759 2795241 0529300
332454117
6677863
7207827 2792173 0529964 3322137
16
45
6680265 9469372 7210892 2789107 0530628
3319735 15
46 6682665 9468707 7213958 2786042 0531293
3317335 | 14
47 6685064
9468042
7217022 2782978 0531958 331493613
48 6687461 9467376 7220085 2779915 0532024
3312539 12
49
668,856 9460710 7223147 2776853 0533790
331014411
50
6672250
9466043 7226207 2773793 0533957
SI 6694942 9465376 7229266 2770734 0534624
3305358
52 6677032 9464708
7232324 2767676 0535292
3302968 8
53 669420
9461040
7235381
2764619 0535900
3300580
7
54 6701307 9463371 77238436 2761564 0536629 3298193
6
55
6704192
9462702 7241490 2758510 0537298
3295808
S
6706576 9462032
7244543 2755457 0537968
3293424 4
57 6708958 9461362 7247595 275 2405 0538638
3291042 3
58
6711338 9460692 7250616 2749354 0539308 3288662
6713716 9460021 9253695 2746305 0539979 3286284
00 6716093 9459342 7256744 2743256
054065 3283907
334625126
28:6749
21
9471364
3326946 18
9470036
II
3307730 / 10
a NO
56
2
و
I
I
46
A Table of Artificial Sines,
28 Degrees.
M
Sines.
61 Degrees.
Tang:
Secants.
I
2
9453960
9
13
14
9447182
9445821
2673473
2670453
7341616
0 9.6716093,9'94593499 7256744 | 10.2743256 10.0540651,01.3283907 60
6718468 9458677 7259701 2740209 0541323 328153252
6720841
9458005
7262837 2737163 0541995
327915958
3
6723213
9457332 7265881
2734119 0542668 327678757
4
6725533 9456659 7268925 2731075 0543341
3274417 56
6727952
9455985 7271967 2728033
5
0544015 3272048 55
6
6730319
9455310 7275008 2724992 0544690 3269681 54
6732684
945 4636 7278048
2721952 0545364
7
326731653
6735047
7281087 2718913 0546040 3264953 52
6737409
9453285 7284124 2715876
0546715 3262.59151
10
6739769
9452609
7287161 2712839 0547391
326023150
6742128
II
9451932 7290196 2709804 0548068 325787249
I 2
6744485 9451255 7293230 2706770 0548745
3255515 48
6746840 9450577 7296263 2703737 0549423 3253160 | 47
6749194 9449899 7299295 2700705 OS SOIOI 3250806 | 46
15
6751546 9449220 7302325 2697675 0550780 3248454 45
I6
6753896 9448541 7305354 2694646 0551459 3246104 | 44
6750245
17
9447862
7308383 2691617 0552138 3243755 43
18
6758592
7311410
2688590
0552818 3241408 42
6760937
19
9446501 7314436
2685564 0553499
323906341
6763281
7317460
2682540
20
0554179 3236719 40
6765623 9445139
21
7320484 2679516 0554861 3234377 39
6767963 9444457
22
7323506
2676494 0555543
3232037 / 38
6770302
7326527
9443775
0556225 3229648 37
23
6772640 9443092 7329547
0556908
24
3227360 36
6774975 9442409
25
7332566
2667434 0557591 3225025 35
26
6777309
9441725 7335584
26644.16
0558275 3222691 34
6779642 9441041
7338601
2661399 0558959
27
3220358 33
28
6781972
9440356
2658384 0559644 3218028 | 32.
6784301
9439671 7344631
2655369 0560329
29
3215699 31
6786629
9438905 7347644
2652356 0561015
3213371 | 30
30
6788955 9438299 7350656
0561701
3211045 | 29
6791279
7353667
2646333
0562388
32
320872128
6793602 9436925 7356677
2643323 0563075 320639827
33
6795923
9436238
7359685
2640315
0563762
34
6798243 9435549 7362693
26,37307
0564451
35
3201757 25
6800560
9434861
7305699 2634301 0565139
3199440 24
6802877 9434172 7368705
0565828
3197123 | 23
37
68051.91
9433482 7371709 2628291 0566558
3194809 | 22
6807504 9432792 7374712 2625288 0567208
39
3192496 | 21
6809816 9432102 7377714 2622286 0567898
3190184 20
40
6812126
9431411 7380715 2619285 0568589 3187874 | 19
6814434
9430720 7383714 26.06286 0569280 318556618
42
6810741
9430028
7386713 2613287 0569972 3183259 17
43
6819046 9429335 7389710 2610290
05 70665
3180954 16
44
6821349
9428643 7392707
260729 0571357 317865115
45
6823651 9427949 7395702 2604298 0572051 317634914
6825952: 9427255
7398696
2001304 0572745
3174048 13
47
6828250 9426561
2598311 0573439
317175012
6830548
7404681
2595319 0574134 3169452 | 11
49
6832843 9425171 7407672 2592328 0574829 3167157 | 10
SO
6335137
9424476 7410462
2589338 0575524 3164863
6837430 9423779
7413650 2586350 0576221 3162570
8
6839720
9423083 7416638 258,3362
05769171
3160280
7
53
6842010
9422386 7419624 2580370 0577014 3157990
6
54
6844297 9421688 9422609 2577391 0578312 3155703
5
55
6846583 9420990 7425594
2574406
0579010 3153417 4
6848868 9420291
7428577
2571423 0579709 3151132 3
57
6851151 9419592 7431559 2508441
0580408
58
24
3148849
6853432 9413893 7434540
2565460
0581107
59
6855712 9418193
60
7437520 2562480
03 81807
2649344
31
9437612
3204077 26
36
2631295
38
41
46
7401689
48
9425806
SI
52
56
I
2
3146568
3144288
Tangents, and Secants.
47
29 Degrees.
60 Degrees.
M
Sine.
Tang.
Secant.
N H
1 23
1. apo
9415388
6869359
TO
II
I 2
6882949
6885209
13
6889723
17
100
TAIN
9404001
}
6714445
01 1 ,
19.6855712 9 9418193 9.7437520 10.2562480 10.0581807, 10.3144288 60
6857991
9417492 7440499 2559501
05 82508 314200959
6360267
9416791 7443476 2556524 0583209 313973358
6862542
9416090 7446453 2553547 0583910 313745857
4
686.816
7449428 2550572
0584012
3135184 | 56
5 6867088
9414685
7452403 2547597
0535315 3132912 55
6
9413982 7455376 2544624 og 86018
3130641 | 54
,7 6871028
9413279 7458349 2541651 0586721
3128372 53
8 6873895
9412575 7461320 2538680 0587425 312610552
19 6876161
9411871 7464290 2535710 0588129 312383951
6878425
9411166
7467259 2532741 0588834 3121575 50
6880688
9410461 7470227 2529773 0589539 3119312 49
9409755 7473194 2526800
0590245 311705148
9409048 7476160
2523840 0590952 3114791 47
14 6887467
9408342 7479125 2520875 0591658 3112533 46
IS
9407634 7482089 2517911 0592366 3110277 45
16
6891978 9406927 7485052 2514948 0593073
3108022 44
6844232 9406219 7488013 2511987 0593781
3105768 43
18 6896484
9405510 7490974 2509026 0594490
3103516 42
19 6898734
7493934 2506066 0595199
3101266 41
20 6900983
9404091 7496892 2503108 0595909 3099017 | 40
21 6903231
9403381 7499850
2500150 0590619
309676939
22
6yo5 476 9402670 7502806
2497194 0597330 3094524 38
23 6707721 9401959 7505762
2494238 0598041 3092279 | 37
24 6909964
9401248
7508716
2491284 0598752 3090036 | 36
25
6912205 9400535 7511669 2488331 0599465
3087795 35
26
93978 23 7514622
2485378 0600177 3085555 34
27 6910683
9399119 7517573
2482427 0600890
308331733
28
6918919
9398396 7520523 2479477
ob0r604 3081081 32
29 6921155
9397682
7523472 2476528 0602318 3078845 31
30 6923388 9396968 7526420 2473580
0603032 3076612 30
6925020 9396253 7529368
24470632 06037+7 30743029
6927851 9395537 7532314 2467666
0604163
307214928
33 6930080 9394321 7535259 2464741 0605179 306992027
34 6932308 9394105
7538203 2461797 0605895 300769236
35 6934534
9393388
7541146
2458854
0600612 306546625
36
6936758 9392671 754408४
2455912
обс7322
3063242 24
37
6738981 9301953 7547029 2452971
0608047 3061019 23
6941203 9.391234
7549969
2450031 0608;66
305879722
39 6943423
9390515 7552908
2147092
0609485 3056577 21
40 6945642 9389796 7555846
244154
0610204 3054358
20
41 6947859
9389076 7558783
2441217
0010)24 305214119
42
6950074
9333356
2438282
NÁ11644 3049926 18
43 6952263 9337635 7564653 2735347
0612365 3047712 17
44 6')54501 9386914 7567587 243241
0613386
3045499 | 16
AS 6956712 9386192 7570520 2429480 0613308
3043288 | IS
46
9.385470 7573452 2426548 0614520 3041078 14
47 6961130 9384747 7576383 2423617 0615253 303887013
48 0903336 9384024 7579313 2420637 0015976 3036654 12
49 6705541 9383300 7582242 2457758
0616700
3034459 II
50
6367745
9.382576 7585170 24114830 0617424 3032255 IO
51
6969947
9381851 7588096
2411704
ObI3149 3030053
69721118 9381126
7591022 2.2009;8
O 16874 3027852 8
53 6.774347
9380400 7593947 2:06033 0617600 3025653
7
5+ 6,76545 9379674 7596871 2410;129 0620326 3023455 6
55
6978741 9370947 7599794 2400200
0621053 3021259 5
50 by+0930
9373:20 7002710 2327284 0621780 3019064
4
6493129 9377402 7605637 234363 0022508 3010871
3
58
6965321 9376764
239 1993 0623236 301.679
59 638751.16 9376035 7611470 2388524 C623905 3012489
00 6,89700
9375305
7614394 2385006 06246;t 3010300
31
32
us as w
38
CO
0
22 ( د ور)
52
20 Nm
57
7608557
2.
I
!
48
A Table of Artificial Sines,
30 Degrees.
59 Degrees.
M
Sine.
Tang
Secant.
O
I
58
2
0628347
4
5
6
7
8
3003742 57
56
299937055
2997198 54
299.501933
299204352
299060051
9
10
II
I 2
13
29798.10 10
14
15
16
2975477 / 44
2973313 43
17
18
19
20
21
22
23
24
25
0641599
0642340
9357660
26
27
28
2.6989700 9.9375 306 9. 7614394
6941387
9374577
7617311
6994073
9373847
7620227
6996258
9373116
7023142
6998441
9372385
7626056
7000622 9371653 7628469
7002802 9370921 7631381
7004981
9370189
7634792
7007158 9369456
763770:
7009334 9363722 7640612
7011508 9367988
7643520
7013681 9367254
7640427
7015852 9366519
7649334
7018022 9365733
7652232
7020190 9365047
7655143
7022357 9364311 7658047
7024523 9363574
7660949
7026687
9362836
7663851
7028842
9362098
7660751
70310II
9361300 7669651
7033170
9360621
7672550
7035329
9359881
7675440
7037486
9359141
7678344
7039641
9358401
76812410
7041795
7684135
7043947
9356918
7687024
7046099
9356177
7689922
7048248 9355434
7692814
7050397
9354691
7695705
7052543
9.353948
7598596
7054689 9353204 7701485
7056833 9352459 7704373
7053975
9351715 770,261
7061116 9350969 7710147
7003256 9350223 7713033
7065394
9349477 7715917
2067531 9348730 7718801
7009667
9347983 7721084
7071801 9347235 7724566
7073933
9346486
7727447
7076064 9345738
7730327
7078194
9344988 7733206
7080323
9344238
7736084
7002450
9343488 7738961
7084575
9342737 7741838
7086092
9341986
7744713
7060022
9341234 7747588
7090943
9340402
10 2385606 10.0624694 10 3010300
00
2382689 0625423
300611359
2379773 0620153 3005927
2376858
062688+
2373944 0627615 3001554
2371031
2368119 0623079
2365208 0629811
2362298 0630544
2359388 0631278
2350480
0632012
298849250
2353573 0632746 2986319 49
2350666
0033481
2984148 48
2347761
0634217
2481978 47
2344857 0634953
2341953 0635689
2y77643 | 45
2339051 0636426
2336149 0637164
2333249 0637902
29711SI 474
2330347
0638040
296898941
2327450
0639379
2960830 140
2324552
0640119
296467139
2321056
0640859
2462514 38
2318760
2960359 37
2315865
2958205 / 36
2312971
0643082
295605335
2310078
0643823
2953901 344
2307186
0644566
295175233
2304295
064530)
294960332
2301404 0646052
2947457 31
2298515 0646796
294531130
2295027
0647541
2943167 29
2292739
0648285 2941025
28
2289853 0649031
293878427
2286967
0649777
2936744
2284083 0650523
2934606 25
2281199 0651270
29.32467 24
2278316 0652017
293031323
2275434 0652765
24126199 22
2272553 0953514
2426007 | 21
2269673 0654262
27239;6 | 20
0655012 22100619
2263916 0655762
2y1967718
2261039 0050512
2717550117
2258102 0657263 2015 42,5
16
225.5287 0658014
20130T 15
2252412 0650766 201117814
2249538
0659518 2909057 113
22.46066
0660271
2906,37 12
2243794
0661024 2404818
2240923
0661778 2002701
2238053 06625,33 2900585 9
223510
0663287 2046471
8
2232315
0664043 28963.5817
2229448
0664709 28974247
6
2226582
0665555 5 2892137 5
2223716
0666312 2840028 4
2220851 0607069 28837920
3
2217988 0667827 2885014
2215125 0668585 2003710
2212263 *0669344 2881607
go
31
32
3.3
34
35
36
37
26
2266794
39
40
41
42
43
44
45
46
ΙΟ
7750462
7093063 9339720 7753334
7095182
9330976
2756200
7047299
9338222
7752077
7029415
9337467
7761!)47
7101529 9336713 7764816
“103642 1335957 7767685
7105753
9335201
7770552
7107303 9334445 7773418
7109972
2333688
176284
7112080
9332931 7779149
9332173 7782012
7116290
9331415 7784875
7118393 9330656
7707737
47
48
49
50
SI
52
53
54
IS
36
57
58
59
60
7114186
2
I
Tangents and Secants.
49
w
31 Degrees.
Sine.
58 Degrees.
3
NI
Tang
Secant.
M
0
I
2
9327616
9326092
0679254
0681553
3
4
5
6
7
8
9
10
II
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
3.3
34
35
36
37
30
39
40
41
42
43
44
45
40
47
48
49
50
SI
52
53
54
55
56
57
58
59
60
9.7113393 | 993300569 7787737 | 10.2212263 | 10.066)344 10.2881607 60
:
7120495
9329897
7790599 2209401
0670103 2879505 59
7122596 9329137 7793459
2206541 0670863 2877404 58
7124645
9328376
7796318 2203682 0671624 2875305 57
7126792
7799177 2200023 0672384 2873208 56
7128889
9320854
7802034 2197966 0573146 287111155
7130983
7804891 2195109 0673908 286901754
7133077 9325330
7007747 2142253 0674670 28669231 53
7135169 9324567 7810602
210)398 0675433
2864831 52
7137260
2323804
7813456 2186544 0676196 2862740151
7139349 9323040 7816309 2103691 0676960 2860651 | 50
7141437
9322276 7819162 2130638 0677724
2858563/ 49
7143.524 9321511 7822013 2177287
0678489
2856476 1 48
714560) 9320746
7824867 2175136
285439147
7147693
9319980
7827713 2172287 0680020 2852307 | 46
7149776 9359213 7330562 2109438 0680787 2850224 45
7151857 9318447 7833410 2106590
2848143 | 44
7153937 9317679 7836258 2163742 0682321 2846063 43
7156015 9316911
I
7839104 2160896 0683089 284398542
7158092 9316143 7841949 2158051 0683857 2841908141
7100168 9315374 78447')4 2155206 0604026 2839832 40
7162243
9.314005
78.476,38 2152362
0685395
283775739
7164336 9313835 78504I 2149519 0636165 2835684 38
7166387 9313065 7853323
2146677 0636935 283361337
7168458
9312294
7856164 2113836 0687706 2831542 | 36
7170526 9311522
7859004 2140996 0688478
2829474 | 35
7172594 9310750 7861844 2138156 0689250 2827406 34
7174660 9309278 7864682 2135318 0690022 282534033
7176725 9309205 7867520 2132480 0630795 2323275 | 32
7178709 9303132 7870357 2129643 0691368
2821211 31
7180351 93076;3
7873193
2126807 0692342 2819149 30
7782912 9306833 7876028
212.3972
0693117 281708829
7194371 9.306109 7878363
0693.91 2815029 | 28
7187030 9305333 7881696 2118304 0694667 281297027
7189086 9304557
7884529 211; 471
o695 4-43
2810914 26
7191142
930376
7837361 23 2139 0690219 2808858 25
719,3196
I 932300- 78.99192 2109808 0696996 2806804 | 24
71952-19 9.;02226
790231 2100977 0697774 280475123
719 7300
9013 789;8/21 2104148 0698552 280270022
7129350 9.200610 7898631
0693330 2800050 21
7201392
7901508
2098792. 07CC109 279810120
7 20341+7 9229112 7904,335 2095615 0700383 279655319
7 205493 9298332 7907IỚI
2092839
0701668
2794507 | 18
7207538 9297551 770947 2090013 070241-19 279216217
7209581 9296770 7912011
2007189 0703230 2790419 1 16
7211623 9295959 791,56,35 2004365 0704011 278837715
7213664 9295207 7918.458
2081542
0704793 2786336 | 14
7215704 9294424 7921200 2078720 0705576
278422613
7217742 9293041 7924101
2075899 0706359 278225812
7269779 9292857 7926921 2073979 0707143 2780121 II
7221314 9242073 7929741
2070259 0707927 277818610
7223848 9241289 7932560
2007-740 0708711 2776152
7225881
9290504
7935378
2004622 0709406 277119
7227913 9289714 79105
2001:05
0710282 2772087 7
7229443
9288,32
79110TI
20;8989
0711068 2770057 6
7231472
9289145
7943027
2056173
0711855 27680.SI
5
723-7000 9287358
7940641
2053359
0712642 2;00000
4
7235026
9.86571 7949455
2050545
0713129 2763974 3
7238051 9285783
79522603
2047732
0714217
2761949
7240075 728499.
7955081
2044919
0715006 2759925
72.12097' 9281205 7957092 20.12108
0715795 2757903
E
2121137
2101312
,
I
(5 بره (926
Q O
IO
on a faced o ano NO in tem a found o
2
I
50
A Table of Artificial Sines,
32 Degrees.
M
Sines.
57 Degrees.
Tang.
Secants.
M
I
2
3
4
5
6
7
8
9
IO
II
I 2
13
14
15
16
17
18
19
20
8013957
21
0732486
22
23
241
25
28
2.7
28
29
go
9.7242097 : 9.9234205 19.7957892 | 10 2042108 10.0715795 10.2757903 60
7 244118
9283415 7960703 2039297
0716585 2755880 i 59
7246138 9282625 7063513 2033487
0717375
275386258
7248156 9281834 7966322 2033078 0718166 275184457
7250174 9281043 7969130 2030870 0718957 2749826 56
7252189 9280251 7971938
2028062 0719749 274781155
72.54204 9279459 7974745 2025255 0720541 2745736 54
7256217 9278666 7977551 2022449 0721334
274378353
7258229
9277873 7980356
2019644 0722127 2741771 | 52
7260210 9277079 7283160
2016840 0722921 273976051
72,62249
9276295 7985964 2014036 0723715 273775150
7264257 9275490 7938767 2011233
0724510 2735743 49
7266264 9274695 7991569
2008431 0725305 273373648
7268269
9273899 7994370 2005630 0726101 273173147
7270273 9273103 7997170 2002830
0726397
2729727 1 46
7272276 9272306 7999:970 2000030 0727694 2727724 | 45
7274278 9271509 8022769 1997231
0728491 2725722 44
7276278 9270711 8005567 1994433 0729289 272372243
7278277 9269913 8008365
1991635 0730087 2721723 42
7280275
9269114 8οΙΙΙ6Ι 1988339 0730886
271972541
7282271 9268314
1936043 0731686
271772940
7284167 9267514 8016752
1983248
2715733 | 39
7286260 9266714 8019546 1980-454 0733286
2713740 38
7288253 9265913 8022340 1977650 0734087
2711747 | 37
7290244 9265112 8025133
1974867 0734888 2709756 | 36
7292234 9264310 8027925 1972075 0735690 2707766 35
7294223 9263507 8030716 1969284 0736493 270577734
7296211 2262704
8033506
1966494 0737296 2703789 33
7298197 9261901 8036290 1963704 0738099 2701803 | 32
7300382 9261096 2039085 1960915 0738904 2699818 31
7302165 9260292 8041873 1958127 0739708 2697835 30
7304148 2250487 80-14661 1955334 0740513 2695852 29
7306129
9258081
80474.17
1952553 0741319 2093871 28
7308109 2257875 E05023)
1949717 0742125 269189127
7310087 9-5706
8053019
1946981 0742931 2689913 26
7312064 2256261 8055803 1944197 0743739 2687936 | 25
731 4040 2255454 8038507 1941413 0744546 2085960 24
7315CIS 9254646
8061370
1938630 0745354 2683985 | 23
7317989 9253837 8064152 1935848 0746163 2682011
7319961 9253023 8056933 1933067 0746972 2680039 | 21
7321932 0252218
8059714
0747732 267806820
7323902 9251408 8072494 1927506 0748592 2676098 | 19
7325870 9230597 8075273 1924,727 072403 267413018
7327837 9249786 8078052 1921943 0750214 2672163 17
7329803 92-18974 8080829 1919171 0751026 2670197 16
7331768 9248161 8083606 1916.294 0751839 266823215
7333731 924739 8086383 1913617 0752651 2666269 14
7335693 9246535 Coo4158 I910842 0753465 2664307 13
7337654 9245721 80419.33 1908067 0754279
2662346 12
7339011 9244907 8092707 1905293 0755093 2660386 II
7341572 9244092 8097480 1:902520 0755908 2658428 10
7343529 9243277
8100253 169.9747 0756723 2656471 9
7345465 9242461 8103025 1096975 0757539 2654515
8
7347440 9241644 8105706
18942,04
0750356 2652560
7
7349393 9240827 8108560 12914344
४
0759173 2650607 6
7351345 9240010
8111336 1888664 0759990 2648655
7353296 9239191
8ΙΙΑΙΟ5
1885895 0760809 2646704 4
7355246 9238373
8116873 1883127 0761627 2644754 3
7357195 9237554
8119641
1880359
0762446 2642805
7359142 9236734 8122403 1877592 0763266 2640858
7361088 9235914 8125174 18748201 0764086 2638912
31
در دی در
32
33
34
35
36
37
22
38
39
40
1930286
41
42
43
44
45
46 6
47
48
49
50
SI
52
53
54
។
ao Noing a homel O
55
56
57
58
5
59
2
I
GO
Tangents, and Secants.
51
33 Degrees.
56 Degrees.
M
Sine.
Tang-
Secant.
M
I
8127939
( 6 )
O O O vaen
52
51
49
I4
20
7399748
26
01 9.736108819.9235914 9.81251741 10 1874826 10.0764086 , 10.2638912 60
7363032 9235093
1872061 0-964907 2636968 59
2 7364976
9234272
8130704
1869296 0765728 2635024 58
3
7366918
9233450 8133468 1866532 0766556 2633082 57
4 7368859
9232528
8136231 1863769 0767372 263114156
5 7370799 9231805
8138993
1861007 0768195 2629201 55
6
7372737 9230482
8141755 1858245 0769018 2627263 54
7 7374675
9230158
8144516 1855484 0769842 2625325 53
8 7376611
9229334
8147277
1852723
0770566 2623389
9 7378546 9228509 8150036 1849964 0771491
2621454
IO 7380479 9227684
8152795 1847205
0772316
261952150
II 7382412 9226858
8155554
1844446 0773142 2617588
12 738-343
9226032
8158311 1841689
0773968
2615657 48
13
7386273 9225205
8161068 1838932 0774795 2613727 47
7388201
9224377
8163824 1836176
0775623 2611799 46
IS 7390129 9223549
8166580 1833420 0776451 260987145
16
7392035 9222721
8169335
1830665 0777279
260794544
17
7393980 9221891
8172089 182791)
0778109
260602043
*I8
7395904 9221062 8174842 1825158 0778938 260409642
19 7397827 9220232 8177595 1822405 0779768
260217341
9219401
8180347 1819653 0780599 260025240
21 7401668
9218570 8183098 1816902 0781430 259833239
22 7403587
9217738
8185849 1814151 0702202 2526413 38
23 7405505 9216906
8188599
1811401 0783094 259449537
24 7407421 9216073 8191348 1808652 0783427 2592579 36
2.5 *409337 9215240 8194096 1805904 0704760 2590663 35
7411251 9214406
8146844 1803156 0785594 2588749 34
27 7413164
9213572
8199592 I 800408 0786428 2586836 33
28
7415075 9212737 8202338
1797662
0787263 2584925 32
29
7416986
9211902 8205084
1794916
0788098 2583014 31
7418895 9211066 8207829
1792171 0788934 2581105 | 30
7420803
9210229
1789426 0789771 2579197 29
7422710 9209393
8213317
1780683 0790007
257729028
33 7424616 9208555 8216060
1703940 0791445 2575384 27
34 7426520
9207717
8218803 1781197 0792283
2573480 26
35
7428423 9206878
8221545
1778455 0793122 25715 77 1 25
7430325 9206039 8224286
1775714
0793961
2569675 24
37 7432226
9205200
8227026
1772974 0794800 2567774 | 23
7434126 9204360 8229766
1770234 0795640 2365874 | 22
४
39 7436024
9203519 8232505 1767495 0790481 2563976 21
40
7437921 9202678 82352414 1764756 0797322 2562079 1 20
41 7439817 9201336 8237981 1762019 0720164
2560183 : 19
42
7441712 9200994
175928: 0799006 2558288
43
7443609
92001SI
1750545 0739849 2556394 17
44 7445408
9199308
8240191 1753809 0700692
2554502 | 16
45
7447390 9198404
8248926 1751074 0801336 2552610 15
7449280
8251660 1748340 0802381
2550-2014
14.5169 9196775 8254394
0803225 254883113
74.53056 9195929 8257127 1752873 0804071 2546944 12
I
49 7454943 9195083 8259860 1740140 0804917
2545057 1 II
50 7456828 9191237
8262592
1737403 0805703
2543172 10
51 7458712 9193390
8265323 1734677 OS06610
୨
7460595 919?542
8268053 1731947
0807458 2539405 8
53
7462477
9191694
8270783 1729217
0808306 2537523 7
54 746.4358 9190845 8273513 1726487 0809155
2535042
6
55 7466237 9189996 8270241
1723759
0810004
25,33763
7468115 9189146
8278969
1721031 OU10854
2531885
4
7469992 9188296 82816y6 171830.1 0811704 2530008
3
7471808
9187445
82.4423
1715577
OS12555 2528132
59 7473743 9186594 8287149
1712851
0813400
2526257
7475617
9185742
828987+
1710126 0814258
2524283
30
31
32
8210574
36
38
I
I
8240719
8243455
9197619
46
47
48
1745600
2541288
52
.
5
56
57
58
I
2
其
​60
52
A Table of Artificial Sines,
I
2
Qend w Nhanel
IL
9175478
I3
IS
I!
ao
7516538
27
34 Degrees.
55 Degrees.
M
Sine.
Tang.
Secant. M
O
9.7475617 19.9185742 9.8289874 10.1710126 10 0814258 | 10:2524383 60
7477489 9184890 8292599 1707401
0815110 252251152
7479360 9184037 8295323 1704677 0815963 2520640 58
3
7481230 9183183 8298047 17019.53
0816817 251877057
7483099 9182329 8300769 1699231 0817671
2516901 56.
5
7484907 9181475 8303492 1696508
0818525
251503355
6
7486833 9180620
8306213 1693787 0819380 2513167 54
7488698 9179764
8308934 1691006 0820:36 251130253
8
7490562 9178908 8311654 1680346 0821092 2509438 52
୨
7492425 9178051
8314374 1685626 0821149
25075 7551
IO
7494287 9177194 8317093 1682907 0822800
2505713 | 50
7496148 9176336
8319811 IÓno 189 0823664 2503852 +9
12
7498007
8322529 1677471 0824522
2501993 48
7499800 9174619 8325246
1674754 0825381 250013447
14
7501723 9173700 8327963 1672037 0320240 249827746
7503579 9172900 8330679 166.)3 2 1 0227100 2496421 | 45
16
7505434 9172040
8333394
1660606 0827960 249456644
17 7507287 9171179 8336109 16638)1 0828821 2492713 43
7509140 9170317 8330823 1001177
0829683 2490260 42
19 7510991 9169455 8341536 1653464
0830545
2489009 41
20
75128.12
9108593 8344219 Ị655751 0831407 2487158 40
21
7514691
9167730 8346901 1053032 0832270 2485309 39
22
9160866 8349673 1650327 0833134 2483462 38
23
7518385 9166002
8352384 1647616 0833998 2448101537
24
7520231 9165137 8355094 1644906 0834,863 2479769 36
5
7522075 9164272 8357804 1642196 0835728 2477925 | 35
26
7523919 9163400 8360513 1632187 0836594 24760%I 34
7525761
9162539 8363221 1636779 0837461 247423933
28
7527602
9161673 8365929 1634071 0838327 2472398 32
29
7529442 9160805 8368636 1631364 0839195 247055831
30
7531280 91599 7 8371313 1628057 084.0003 246372039
7533118 9159069
3374042 1625951 0840931 2466882 29
7534954 9158200
8376755 1623245 0841800 246504628
33 7536790 9157330 8379460 1620510 0342670 246321027
34 7538624
8302164 1617836 0843540
2461376 | 26
35
7540457
8384867
0344411 2459543 25
7542288
9154718
8387571
0845282
245771224
7544119 9153840
8390273 1002727 0846154 2445588123
7545949 9152474 8342975
1607025
09.17026 2454051 22
7547777 9152101 8395676
J604324 0747899 2452223
49
7549604
8390377 1001023
245039620
7551431 2150354
8.401077 1593923 0849646 2448509 19
755,3256
9119479 810.3776 159022.4 0%50521 2446744 | 18
43
914860H 8400475 159.3525 0851396
244492017
44
7556902 9147729 8404174
0852271
2443018 16
45
7558724 9140852
8411871 1988129 095.148 2441276 15
7560544
9145976 8414569
1585-131 0054024
2439456 14
47 756 2364 9145099 8457265 1582735 0854201
2437636 13
48
914-1221
2419961 1580039 0855779
243581812
49 7565999 914.3342 8422657 IS 77.343 0856658 2434001
50 7567815 9142404
8425351 1574042 0857536 2432185 10
51
7569630
9141581 8428046 157.1954 0856416 2430370 9
7571444 9140701
8430732 156926Ị i 0359296 2428556
7573256 6
-4 84334.32 Ig60563 0800176 2426744 7
ñ 4 7575063 913294,3
8436125 156,3875 0861057 2424932
6
7376878
8458817 TS
2423122 5
7578687 9137179 8441508 15504921 0862821
2421313 4.
57
7580495
9136296
8444192
1555801 0803704 2419505 3
50
47582302 9135419 8146889 1553111 0204587 2417698 20
7534 Iof 91345.30
8419579
1550421
0865410 2.415892
60 7585913 9133645 8952208
1547734
2414087
31
32
9150460
9155589
161:133
161 2429
36
37
38
39
21
9151228
0878772
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42
7555080
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40
7.564182
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52
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213601
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35
56
0363939
3 ๆ ๆ ๆ "
O O Cc an fan
99
I
0866355
Tangents, and Secants.
53
35 Degrees.
MI
Sines.
54 Degrees.
Tang.
Secant.
M
I
1
2
4
va
Ιο
II
I 2
8484.492
8437174
If I TH 4
13
IA
15
ame
17
20
21
22
23
0 2.7585913 9.9133645 9 8452263 10.1547732
7587717 0132760 84549561 'I545044
7589519: 9131875 8457644
1542356
3 7591321 9130989
8400332
1539603
7593121 9130102 8463018
1536982
5
7594920 9129215
8405705
1534295
6 7596718 9128328
8468390
1531610
7 7593515 9127440
8471075 1528925
8 7600311 9126551
7473760 1526240
9
7602106 9125662 8476444
1523556
9603892
9124772
8479127 1520873
7605692
9123882
8181810 1518190
7607483 9122991
1515508
7609274 9122099
1512826
7611063 9121207 8489855 1510145
7612851 9120315 8492536 1507464
16
7614638 9119422 8495216 1504784
7616424 9118528 8497896 1502104
18 7018208 9117634
8500575 1499425
19 7619992 9116739 8503253 1496747
7621775
9115844
8505931 1424069
7623556
9114948
8508608 1121392
7625337 Q114051 8511285
1488715
7627116 9113155 8513961 1436032
21 7628894 9112257
8516637 1483363
25 7030671 9III352 8519312 1480688
26 7632447 9110400 8521987 1478013
27 7034222
9109561
8524661 1475339
28 7635996 9108661
8527335 1472665
22
7637769 9107761
8530008 1469992
7639540 9106860 8532680 1467320
31 76413II 9105959 8535352
1464648
32 7643080
9105057 8538023 1461977
33 7044849 2104155 8540094 1459306
3- 7646016
9103251 854.3365 1450635
35 7648382 DI02348 8546034 145,3966
7650147 QI014+4 8548704 1451296
37
765I9II 9120539 8551372 1448628
7653674 9099634 8554041 1445959
39 5655436 2098728 8556708 I443292
40
7657197
9097821 8559376 1440624
41
7650957 9096915 8562042 1437958
7660715 9096007 8564705 1435292
43 7662473 Doyog9
8567574 1432626
44. 7664229 9094190
8570039
1429961
4,5 7665985 9093281
8572704 1.427296
7667739
9092371 8575368 1421632
7061492 9011461
8578031 1121969
48
7671244
90905.50 8530694 ILI9306
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14 Τύ643
7674746 9088727 8586019 1413981
7676494 9087814 8588080 1411,20
7678242 9086901 8501341 1408659
53
7679989
9085988 4594002 1405998
54
7681735 9085073 8596061 1402,339
55
7683480 908.1159 8599321 1400679
56
7685223 9083743 8601980 1398020
57 7636966 9082327 8604638 1395.362
7688707 9081411 8607296 1392704
59 7690418
9080494 8609954 1390046
76921871 9079576 8012010 1387390
30
10.0866355 , 10.241408760
0867240 241228359
0868125 241048158
0869011
24,08679 57
0869898 240687956
0870785 240508055
0871672 2403282 54
0872560
2401 485 153
0873449
2399689 ( 52
0874338 2397894 i 51
0875228 239610150
0876118 3394308 49
0877009
2392517 48
0877901 2390726 47
0878793 2388937 46
0879685 2387149 45
0880578 2385362 44
0881472
2383576 43
0882366 2381792 42
0883261
2380008 41
0884156 2378225 40
0885052 2376444 39
0885949 2374663 38
0886845 2372884 37
0887743
2371106 36
0888641 236932935
0889540
2367553 34
0890439 236577833
0891339
2364004 / 32
0892239
236223131
0893140 236046030
0894041 2358689 29
0894943 23569201.28
0895845 235515I 27
0896749
235338,426
0897652
2351618 25
0698556
237985324
0899461 234808923
0900360 2346326 22
0901272 2344564 21
0902179 231280320
OYO3085 2341043 | 19
0903993 2339285 18
O90.1901 2337527 | 17
0905010
2335771 | 16
0906719 2331015 15
0907629 23.32261 14
0908539 2330508 13
0909 450 2328756 12
0910361 2317001 II
0911273 2325254 10
0912186
2323500 9
0913099
2321758 8
0911012
2320011
7
0917927 2318205 6
0915841 2316$20 5
OY16757 2314777 나
​0917673 2351037 3
0918589
2311273
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09 20424 2307813
36
3.8
42
46
47
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50
SI
52
58
2
I
ܘܹ
54
A Table of Artificial Sines,
36 Degrees.
M Sine.
53 Degrees.
Tang.
Secant.
M
I
8617923
4
II
دي دي
17
1339650
III H 22
я
оло
23
0 9.7692187 19.9079576 9.86 12610 , 10.1387390 10.0920424 , 10.2307813 60
7693925 9078058
8615267
1384733
0921342 230607559
7695662
9077740
1332077 0922260 2304338 58
3 7597398 9076820 8620578 1379422 0923180
2302602 57
7699134 9075901 8623233 1376767 0924099 2300866 56
5
7700868 9074980
8625887 1374113 0925020 229913255
6 7702601 9074059
8628541 1371459 0925941 229739954
7
7704332 9073138 8631195 1368805 0926862 229506853
8 7706063 9072216 8633898 1306152 0927784 229393752
9
7707793 9071293
8636:00 1363500 0928707 229220751
Το 7709522 9070370
8639152 136084.8 0929630 229047850
7711249 9069446 8641803 1358197 0930554
228875142
I 2 7712976 9008522
8644454
1355546 0931478 2287024 48
13 7714702 9067597
8647105 1352895 0932403 2285298 47
14
7716426 9066671 8649755 1350245 0933329 228357446
15
7718150
9065745
8652404 1347596 0934255 228185045
16 7719872 9064819 8655053 1344947
0935181
2280128 | 44
7721593 2063892 8657702
1342298 0936108
2278407 43
18 7723314 9062964 8660350
0937036
2276686 42
19 7725033 9062036
866297
1337003 0937964
2274967 41
20 7726751 9061107
8665644
1334356 0938893 2273249 40.
21
7728468 9060177
8663291 1331709
0939823
2271532 32
22 7730185 9059247
8670937
1329063 0940753 2269815 | 38
7731900 9058317 8673583 1326417 0941683
2268100 | 37
24
7733614 9057386 8676228 1323772 0942614
2266386 36
25 7735327 9056454 8678873 1321127 0943546
2264673 | 35
26 7737039 9055522
8681517
1318483 0944478
226296134
27
7738749 9054589 8684160 1315840
0945411
226125133
28 7740459 9053656 8686804
1313196 0946344
2259541 32
29
7742168
9052722 8689446 1310554
0947278
225783231
30
9051787 8692089 1307911 0948213
2256124 30
31 7745583
9050852 86947
3
1305269 0949148
2254417 29
7747283
9049916 8697372 1302628 0950084
225271228
33 7748993
9048980 8700013 1299987
2251007 27
34
7750697 9048043 8702653 1297347 0951957
2249303 1 26
35 7752399 9047100
8705293 I294707 0952894 2247601 i 25
36 775 4101 9046168 8707933 1292067
0953832
2245899 24
37
9045230 8710572 1209428 0954770 2244199 23
38 7757501 9044291
8713210 1286790 0955709 2242499 22
7759199
39
9043351 8715848
1284152
0956649 2240801 21
40
7700897 9042411 8718486
0957589
2239103 20
7762593 9041470 8721123 1278877
41
0958530
2237407 19
7764289
9040529
8723760
42
1276240 0959471
2235711 18
7765983 9039587 8726396 1273604
43
09.04.13 2224017 17
7767676
9038644 3729032 1 270968 0961350 2232324 16
44
7709369
9037701 8731008
0962299
45
2230631 15
7771060
9036757
46
8734302
1265698 0963243 2228940 14
7772750 9035813
8730937
1263063 0964187
4.7
2227250 13
7774439
48
9034808
8739571
I 260429
0905132 2225561.12
7776128
9033923 8742204 1257790 0966077 2223872'11
49
7777815
9032977
8744838 125,5162 0967023 2222185 10
7779501 9032031
8747470 1252530 0467969
51
2220499
7781180
9031684 8750102 1 249898 0908916
2218814 8
52
7782870 9030136
8752734
1247266 0969864 2217130 7
53
7784553 9029188
87:5305 1244635
0970812
2215447
6
7786235
9028239 8757996 1242004
2213765 5
55
7787916
8760627
9027289
1239373
0972711
56
7212084 4
7784596
9026339
8763257
1236743
2210404 3
7791275 9025389 8765866 1234114
2208725
7792953 9024438 8700515 1231405
2207047
59
60 77946301 9023486
87711644
1220056
09765241
2205370
7743876
www
32
0951020
7755801
1281514
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1268332
so
54
0971701
I
0973661
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57
58
2
I
Tangents, and Secants.
55
37 Degrees.
Sine.
52 Degrees.
Secant. M
M
Tang.
O
I
9021581
9013938
218865650
II
2180336 | 45
2177016 43
2175357 42
8821007
21
7831268
2168732 | 38
2165425 36
216377335
27
28
2160472 33
در دم دری بی بی بی
30
31
32
9.7794630, 9. 90234869.8771144 , 10.1228856 / 10.0970514 : 10.2205370 60
7796306
9022534 8773772
I 226226
0977466
220369459
7797981
8770400 1223000 0978419 2202019 58
3
7799055 9020628
8779027 I 220973 0979372 2200345 57
4
7801328 9019674 8781054
1218346 0980326 2198672 56
S 7803000 9018719 8784281 1215719 0981281 219700055
61
7804671 9017764
8786907 1213093 0982236 2195329 54
7800341 9016808 8789533
I210407 0983192 219365953
8
7808010 9015852 87)2158
1207842 0984148 219199052
9 7809677 9014095 8744732 1205218 0985105 219032351
10 7811 344
8797407 1202593 0986062
7813010 9012980
8600031 1199969 0987020
218699049
I2 7514675 9012021
8802654 1197346
0987979 218532548
13 7816339 2011062 8805277 I194723 0988938 2183661
47
14 7818002 2010102 8807900
IIY2100
0989898 2181998 46
15 7819664 9009142
8810522
1189478 0990858
16
7821324 9006181 8813144
1180856
091819 2178676
44
I7 7822984 9007219
8815765
1184235
0992781
18
7824643 9006257 8818386
II81614
0993743
19 7826301 9005294
1178993
0994706
217309941
20
7827958 9004331
8823627
1176373 0495669
2172042
40
7829014 9003367 8820246 1173754 0996633 2170386
39
22
9002403
8623866 1171134 0997597
23 7832922 9001430 8331484
1168516 0998562 2167078
37
24
7834575
9000472
8834103
1165897
0999528
25
7836227 8999506
8836721
1163279 1000494
26
7837878 8998539 8839338 I160602
1001461
2162122
34
7839528 8997572 8841956 1158044 1002428
7841177 8996604
8844572
1155428
1003396 2158823
3.
29 7842824 8995636
8847189
1152011 1004364 2157176
3I
7844471 8994607 8849805 IIS0195 I005333
7846117 8993697
3852420
I 147560
1006303
2153883
29
7847762 8992727 8855035 1144905 1007271 2152238 28
33 7849406 8998755
8857650 1112350 1008244
34
7851049 8990784 8860264 1139736 1009216
21.7895126
35 7852691 8989812 8862878 1137122 IOIOIS
7854332 8938840
8865492 11.44503 IOI1160
37
7855972 8987867 886310S IIgIv95
101213,3 2144028
38
7857611 8986893 8870718 1129282 IOI3107 2142389
39 7859249 8985919 8873330 1120070 IOI4081
2140751
40
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8984944
8875942
1124053 1015056
2139114
7862522 8983968 8878554 II21446
1016932 2137478
19
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7864.157
8982992 8281165 IT10835 101;008
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2155529 30
215059427
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2147309 25
2145668 24
23
22
21
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A Table of Artificial Sines,
38 Degrees.
M
Sine.
51 Degrees.
Tang.
Secant.
M
1
I
2.
Ovaan
7917566
7912168
7920769
7922362
0 17.7893420 | 9.8965321 9.8928098 10.1071902 10.1034679, 10 2106580 60
7895036
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1035666 2104964 52
7896652 8963346 8933306
1066694
1036654 2103348 58
3
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8962358 8735709
1064091
1037642 2101734 57
4 7899880
8961369 8938511
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1038631 2100120 56
5 7901493 8960379 8940114 1058886 1039621 2098507:55
6
7903104 8959389 8943715 1056285 1040611 209689654
7 7904715 8958398 8946317 1053683 10410OZ 2095285:53
8 7906325 8957406 8948918 1051082 10.12594
2093675 152
9 79079.3.3 8956414 8951519 1048481 1043586 209205751
IO 7909541 8955422 8954119 1075881 1044578 209045950
II 7911148
8954422 8956719 1043281 1045571
2088852 49
I 2 7012754 8953435 8959:319 1040681 1040505 2087246 48
I3 7914359
8952440 8901913
1038082 1047560 2085641 | 47
14 7915963
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16
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2030332 44
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19 7923968 8946461
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20 7925566
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21 7927163
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22 7928760
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23
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24 7931949 8941462 8990487 1009513 1058538 20680511 36
25 7933543 8940461 8993082 1006918 1059539
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26
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7936727
27
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28 7938317 8937452 9000865 0999135
1062548 2061683 32
7939907
29
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1063552
0926541
2060093 31
7941496
30
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7944670
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2052159 26
7949425
25
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7952520
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1072615
0973214
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106.7
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2039,514 18
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44
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57
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Tangents, and Secants.
57
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A Table of Artificial Sines,
40 Degrees.
49 Degrees.
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9 8080675 9.8342546 2.9238135 110.0761805 10 1157460 : 10.1919325
60
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19 8109121 8822285 9286835 0713165 1177715 1890879 41
20 EIT0009 8821213 928)396 0710604 1178787 1839391 40
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23
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24
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25 8118033 8815842
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26 8119521 8814766
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42
8147534
0631341
8785470
57
58
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I
Tangents and Secants.
59
41 Degrees.
48 Degrees.
Secant. M
M
M
Sine.
Tang.
I
8774501
9409486
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tmt #11 1
20
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52
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Ho anco o N y to
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60
A Table of Artificial Sines,
42 Degrees.
47 Degrees.
M
Sine.
Tang.
Secant.
M
1
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2
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I
20
I
เ
26
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28
8679779
+
098255109 7.8710735 1 9.9544374 10.0455026 10.1289205 { 10.1744891 vo
8256512 8709597
95+6915 045 3085 1290403 174348859
8257913 8708458 9549455 0450545 I291542
1742087 58
3
8259314 87073 9
9551995 0448005 1 242681 740636 57
4
8260715 8706179
9554535 0445465
1293821
1739285 56
5
8 62114 8705039 9557075 0442925 1294961 1737886 55
6
8263512
8703898 9559615 0440385 1296102 1736188 54
8264910 8702756 9502154 0437846
1207244
1735690 53
8
8701613 9564694
0435306
1293387
1733093 52
9
8267703 8700470 9567233 0432767
129530 173229751
IO
8269098 8699326 9509772 0430228
1300674
173090250
II
8270493 8698182 9572311
0427689 I301818 1749507 1 49
827188/ 8697037 9574850 0425150 1302963 1728113 : 43
13
8273279 8695891 9577389 0422611 1304109 1726721 42
It
8274671 8694744 9579927 0420073 1305256 1725329:46
IS
8276063 8693597
9502465 0417535 1300403 1723937 | 45
8277453 8692442
9585004 0414996 14307551 172254744
17 82;8843 8631301 9587542 0412458 1308699 1721157 43
18
8280231 8690152
95400oo
0409920 1309848 1719769 42
19
828161) 8689002
9592618 0407.382 1310998 1718381 41
6283006
9575155 0404845 1312149 1716494 40
8284393 8636700 9507693 0402307 1313300 1715697 39
22
8285778
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8685548 9600230 0399770 1314452
171422238
8287163
23
8684396
9602767 0397233 1315604 1712837 37
24
8288577 868 242 9605305
0393905 1310758 1711453 36
25
8289930
8682088 9607842
0392158
1317912 1710070 35
8291322
8680934 2610378 0389622 1319066 1708688 34
8292024
9612915 0387085 1320221 1707306 33
8294075
8678023 9015452 0384548 1321377 170592532
22
8295454 8677466 9617988 0382612 1322534 " 170454031
8296833
8676309 0620525
30
0379475 1323691 170316730
8298212 8675151 9623061 0376939 1324849 1701788 29
8299589
8673992 9625597
0374403 1326008
1700411 28
8300gbo 8672833 9628133 0371867 1327167 169903427
8322342 8671073 9630669 0369331 1328327
1697658 26
8303717 8670512 9633204
0366796 1329488
35
1646283 25
8305091 8662351 2635740 0364200 13300449 169490424
8306164 8668189
9634275
0361725
13318II 1093536 23
8301837 8667021 9640811
0359189
1332974 1692103 22
8;29209 8665863 9643.346 0356054
39
1334137 1690791/21
8310580
8601699
9645381 0354119 1335301
168942020
8311950
8663534
9648416
0.35158.1
1336.06 108805019
8313320 3662369 y050951 0349049 1337631 1680680 18
8314658 8661203 2053486 03465524 13,8797
43
1685312 17
8316CSC 8600036 9056020 0343480 1339904
44
1683944 ( 16
3317-3 8658868 9058535 0341445 1341132 1602577 | 15
8313769 8657700 yibiosy 0338911 13 2300
1681211 14
8320155 650531 9663623
47
0336577 1343469 1677845 I3
021S E651352
9666157
0353843 1344633 1670481 12
(1322883 865a 142 36686)2
0331,06 1345008
49
1677117 II
932.,230 8053021 9672225
0328775
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1340979 167575410
8325609 8651849 2673759
0;20241 1348151
1674391
9
8326470 8050077 9676293
52
0323707 1393-3 1673030
8
8328331
53
8049504 0678827
032 1173 1350.196 1671669 7
832460 I 8648331 9061300 0313610 1351669 1670309
6
54
8331050 56756 9003:33 0316 107 1352844 1608950 5
55
8332408 8645431 9650427 OjI3573 13.4019
31
32
33
34
پ ن در به در بر
at mano
36
37
38
1
47
41
42
45
40
43
SI
I
I
1667592
4
83,33766 8644806 9638960 0311040 1355124 160623.4 3
8.335 122 8643929 9691493 0308507 1350371 1664873 i 2
2,
8536478 864-4,52 969-70261 0305974 1357548
1663522
60
8337833 0641275 9696559 303441 1398725 1662,167
56
5.7
58
59
1
J
Tangents, and Secants.
61
43 Degrees.
M
Sine.
46 Degrees.
Secant. M
Tang.
I
2
3
4
3636557
Obrand no ao
5
6
863-194
7
8
I 368172
862)400
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12
IC
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13
8625902
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17
18
19
20
21
22
8615IYO
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833918,
86400,16
9699091 0330903 1359904
166081259
8340541 8036917 9601024 0293376 1301083 1659459 58
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8637737
9704157 02958.13 1362263 1658106 57
8343210
970663) 0293311 1363443 1656754 ! 56
8344597 8635 376 9709221
02.077)
1364624 1655403 55
15345248
9711754 0288246
1365806
1654052 54
834727
8633011
9714236
028571 1366989 1652703 53
8348646 8631823
2716818
0283182
1651354 52
83+2294
8130644 9719350 0230650 136:9356 165000651
8351341
9721832
0278118 1370540 164865950
8352688 8626274 9724413 0275507 1371726
1647312 49
*354033 8627033 9726945 0273055 1372912 164596748
8355378
9729477 0270523 1374098 1644622 47.
8350722
8924714
9732008
0267992
1375286 164327846
8358066
8623526 9734534 0265461 1376474 1641934 45
8359408 8622333
9737071
0262924
1377662
1640522 | 44
6360750 8621140 9739602
Q260398 1378852 1639250 43
8362091 861.9958 9742133 0257867 1380042 1637909 142
8363431 8618767
9744664
0255336 1381233 1636569 / 41
8354771 8617576
2747195 0252805 13321244 1035229 40
3366107
8016383
9749726
0250274 1383017 1633891 39
8367447
9752257
0247743 1384810 1632553138
8368784
9754787
0245213 1386003 1631216 37
8370121 8612803
0212682 1387197 1629879 36
837145
8611608 9759879
0240ISI 1388392 1628544 35
8372791 8610412 9762379 0237621 1389588
1627209 34
8374125 8609215 9764909
0235091 1390785
1625875 33
8375458 8608018 9767440
0232500 1391982 1624542 32
$376790 8606821 9769970 0230030 1393179 1623210 31
8378122 8605622 9772500
0227500 1344378
1621878 30
8379453
8604423
9775030 0224970 1395577
1620547 | 29
83807831 8603223
9777500
0222110 1396777 1619217 28
8382112! 8602022
9780090
0219210 1327978
161788827
8383441' 8600821 9762620
0217330
139)172 16165591 26
$38.4769 8599619 9785149 0214851 1400381 161523125
8386096 8593416 9787079 0212321 1401584
I61390424
83674221 8597213 9790209
0209791 1402787 10125732;
83807471 8546oy
9792738
0207202 1403991 I61125322
8390072 8594804
0204732 1405196 16099284 21
6391396: 8593599 9797797 0202203 1406401 1608604 20
8372710 8592393 9000326 0197674 1407007 160728119
8394041: 8590136 9802856 0191144 1408814 160595918
63425263 8587978 9805345 0194015 1410022
160463717
320084 8538770
9807414
0192080 III 230 160331016
8375004 8,87501 90107431 0109557 1712439
1601990 15
831):23, 8:86351 9312:) : 2
OIS:028 1413649 1600677 14
8409012 853;171 9815501;
1414859 159935813
0401159 85 034291 0180301 01919; 1410071 159)804112
94232,6" 8;82710 98205591 C179741 1417282 159672411
8.104593: 3581505 9823087! 0170413 1418.195 154570; TO
IO
8405708, E580292
9825616 0174384 1419708 1594092 9
8.j!)7223! 8579u;8 9828145! 0171895 14.20922 1992777
8
853785371 8977 sing
&
9330673; 016932; 1422137 I541403 7
8.608:
08;70018 1833232 or 0798 1423352 159015C
8410102 0575432 9035730 OIÓ+2; 1424568 1588838
8412471 3574215 9838259 0101741
1425785 1587526
8413755
8372998 79.70787) 0153213 1427002 1500215 3
5747:) 98.+3315 0150635
20231
1584905
8410404 8,70501 96456-Y 0154156
1.129439 1583596
8569341 98483721 OISI628 1+30654 1582287 io
27
28
29
30
31
32
33
34
35
36
37
L
کر دی
33
را در دی دا
9795263
39
40
41
42
43
44
45
46
47
40
49
7
OI 34199
50
SI
Saj
;
52
53
5.4
55
56
$7
58
59
60
8415095
24
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817713
62 A Table of Artificial Sines, ان
. &c.
44 Degrees.
45 Degrees.
M,
Sine,
Tang.
Secant.
M
I
2
13
14
21
22
0090965
0 9.84177139.85693419 9848372. 10.0151028 10 1430659, 10.158228760
8419021 8568121 9850900 0149100 1431879 158097952
8420328 8566900 9853428 0146572 1433100 1579072 58
3
8421634
8565678 9855956 0144044 1434322 1571306:57
4
8422939 8564455 9858484
0141516 1435545 1577061156
5
8424244 8563232 9861012 0138988 1436768 1375756 55
ó 8423548 8562008 9863540 0136460
1437992 157445254
7
8426851 8560784 9866068 0133932
1439216 I573149 53
8428154 8559558 9868596 0131404 1440442 1571846.52
9
8429456 8558332 9871123
0128877
1441668
1570544 51
Ιο 8430757 8557106 9873651
0126349
1442894 1509243 50
II 8432057
8555878 9876179 0123821 1444122
1567943' 49
12
8433356 8554650 9878706
0121294
1445350 1566644 48
8434655 8553421 9881234 0118766
1446579
1565345 47
8435953 8552192 9883761
0116239
1447808 1564047 | 46
IS
8437250 8550961 9886289 0113711 1449039
1562750 | 45
16 8438547 8549730 9888816 ΟΙΙΙ184
1450270 1561453 44
17
8439842 8548499 9891344 0108656 1451501 1560158'43
18 8441137 8547266 9893871 0106129 1452734 155886342
19
8442432 8546033 9896399 0103601 1453967 155756841
20 8443725 8544799 9898926 OIO1074
1455201
1556275 40
8445018 8543564
9901453 0098547
1456436
1554982 39
8446310 8542329 9903981 0096019 1457671 1553690 38
23
8447601
8541093 9900508 0093492 1458907 1552399 37
24
8448891 8539856 9909035
1460144 1551109 / 36
8450181
25
8538619 9911562
0088438 1461381 154981935
26 8451470
8537381
9914089
0085911
1462619 154853034
27
8452758 8536142 9916616 0083384 1463858 1547242 33
28 8454045 8534902 9919143
0080857
1465098 154595532
29
8455332
8533662 9921670
0078330 1466338 1544668 31
8456018
30
8532421 9924197 0075803
1543382 30
31
8457903 8531179 9926724 0073276 1468821 1542097 29
84.59188
32
8529936 9920251 0070749 1470064 1540812 28
8460471
33
8528693
9931778
0068222 1471307 153952927
8461754 8627449
34
9934305 0065695 1472551
1538246 | 26
8463036 8526204
35
9936832 0063163 1473796 1536964 | 25
8464318 8524959 9939359 0060641 1475041 1535682 | 24
37
8465599 6523713
9941886
0058114 1476287 1534401 23
38
8466879
0522466
9944413 0055587 1477534 1533121 22
8468158 8521218 9946940 0053060
39
1478782 1531842 21
8469436 8519970
40
9949406
0050534 1480030 1530564 | 20
8470714 8518721
0048007
9951993
41
1481279 1529286 19
8471991 8517471
42
9954520 0045480
1482529
1528002 | 13
8473267 8516220
43
995 7047 004295-3 1483780 1526733 17
8474543 8514969 9959573 0040427 1485031
1525457 16
8475817 8513717 9962100 0037900 1486283
4
152418315
8-477091 8512465 9964627
40
0025373
1487535 152290914
8478365 8 SII211 9967154 0032346
1488789 1521635 13
47
8479637 8509957
9969630
0030320 I 490043 1520363 12
8480909 8508702
9972 207
0027793 1491298 IS19091 11
II
8482180 8507416 99747.34 0025266
1492554
50
1517820 10
8483450 8506190
0022740
9477260
1493810
51
1516550
8484720 8504933 9979787 0020213 1495067
15152801 8
8485989 8503675 9982314 2017680
1496325 1514011
7
53
8487257
8502417
9944340 0015160
1497583 151274.3
6
54
8488524 8501157 9987367 0012633 1498843 IS114765
55
8489791 %494897 9989093 OOICIO7 I5C0103 1510209 4
1467579
دي ديه به
36
48
49
IO
52
wa COO
56
8491057 8448637 9992420 0007520 I501363 1508943
57
3
8492322 8497375
58
0005053
9994947
1502625 1507678
8490113
2907473 COO2527 1503887 1506414
59
84940501
84948501 OCCOOCO
60
1
0000000 1505 150
Q
1505150
1
8-93536
-
1
G
ASQsa
A
T A B
T
B
L E
OF
NATURAL SIN E S.
ප්‍රද
**
:
:
64
A Table of Natural Sines. .
0.000
م) ور
I
Oliy om
2
5818
8727
10
II
29 089
12
34 906
37 815
13
52 360
55 268
OO
21
22
63 995
66 9n4
75 630
M o Deg. 89 Deg. i Deg. 83 Deg. 2 Deg. 87 Deg: 3 Deg. 86 Deg. M
10300 000 , 174.524 9998 477 348.995 } 9993.908 523.360, 9960295
||
2 909 9999 996 177 432
E 426 351 902
93 806 526 264
$
86 143 59
998 180 341 8 374 3544 809 93 704 529 169) 85 yio 58
3
9y5 183 249 8 321 357 716 93 600 572 074
85 83.5 57
4 II 636 993 16158
207 360 623 93 495534979 85 60 56
$ 14 544
989 189066
8213 1 353 530
93 390 537 : 85524 55.
6
17 453
984 191 974
8 1571 366 437 93 284 5-10779 85 367 54
7 20 362
979 194385
8101 369 344
9,3 177 | 543 693 85 209 53
8
23 271
973 197 791
8 044 372 251 93 069 5465) 8; 050 52
9
26 180 960 200 699 7986 | 375 158 92 960 549 502 84 8,1 51
958 203 608 7 927 378065 92851 552 400 84731 50
31998
949 206 516
7867 380971 92 740 555 31 1 84 570 49
939 209 424 78073 83 873 92 629 558215 87 408 48
928 | 212 332 7 745 386 785 92517 561 119 84 245 47
14
40 724
917 215 241 7 683 389 692 92 404 564024 84021 46
43 633
15
905 218 149
7620 392 598 92 290 560928
8.3 917 45
16
46 542
892 221 057 7 556 395 505 92176 569832 83 751
17
49 451
878 223965 7 491 398 411 92060 572 7,36 83 585 43
18
863 226873 7 4261 401 318 91 944 575 640 83 418.4%
19
847 229 781 7 360 404 224 91 827 578544 83250 41
20
58177 831 232 690 7 242407 13 I 91 7091 501 448 83 082 40
61 086 813 235 598 7 224 410037 91590 584 352 82 912 | 39
795 238 506
7156412 944
91 470587256 82 742 / 38
776 241 414
7086 415850
23
91 350590160
8.2 57037
24
69 813 756 244 322 7015 418757 91 228 593 064 83 398 36
72 721
736 247 230 6 943 421663
25
91 106 595 967 82 225 35
26
713250138
6871424569
90 983598871 82 052 34
78539
691 253 046
6 798 427 475
27
90 859 601 7,5
8187733
28
81 448
668 255 954
6 724 | 430 382
3 90 734 604 678
81 701 | 3%
32
84 357
644 258862 6 649 433 288
29
90 609 607 582 8152531
87 265 619 261 769
6 573 | 436 194
90 482 610 485
30
81 348 30
90 174
6 497 439 100
31
90 355 613 379
81 17029
566 267 585 6 419 442 006 90 227 616 292 86 991 28
32
6 341 444 912
5391 270493
80 8II
90 098 619 196
33
95 992
27
98 900
SI1273 401
6 252 477818
34
89 468 | 622 099
80631 26
IOI 809 482 276 309
35
89 837 625 002 80450 35
104 718
36
452 279 216
6101453630 89 706 627 905
80 267 114
107 627
421 282 124 5020 | 456536 89 573 630 808
37
80084 / 23
38
IIO 535
389 235 032 S 937 459 442 89 440 633 711 79900
113 444
357 287 949 5854467347 89 306 | 636 614
39
79716 21
323 290 847
5770 465253 89 171 639.517
40
79.530 | 20
289 293 755
5684 | 468159 89035 | 642 420
41
79 343 | 19
I 22 170
42
$ 599471 C6
88 899 645 323
218 299 570
125 079
5 512 473970 80761 648226 78 808/17
43
127 987
181 302 478 5424476876
88 623 1651 129
44
78779116
130 896
143 395 385
5336479 781
45
88 484 | 654031 78587115
104 308 293
46 133 805
5247422687
88 344 656 734
14
136 713
065 / 311 200 5157 | 485592 88 203 659836
47
77 207 13
48
025 | 314 108
5036 488498
88 061 662 739
78 015 12
142 530 9998924 317 015 4 974 491 403
49
87919 665 6,11 77821 II
8
145 439
4831 494 308
942 | 319 922
87775 068544
77627.10)
148 348 8 899 322 830 4788 497 214
87631 078 170
I
77433
51
8255 325 737
ISÍ 256
4693 500119
87 486674249
8
77237
52
8 %11328 644 4 598503024
87 34.0 6977.251 770497 7
53
8 766 331 552 4502505929
157 073
87194) 680153 76 843
ó
54
8720 334 459 4405 1508835 87046 633 055 70 645 5
55
162 890
8 673 337 306 4.308 511 740 858981 685957
8 625 | 340 273
4209 | 514645
86 748 688859
3
8577 343 181
4110517550 86.598 691761 70045
8527 346 088
4009 520 455 86 447 | 694 663 75 843
59
2 3
O
593 264673
93 083
61182 450724
w to
22
116 353
119 261
254 296 662
7915618
78399
139 622
50
9
154 165
159 982
56
57
76 445
76245
165 799
168 707
다
​58
I
8 4771 348 995 39081523 360 86 295697 565
60 174 524
171 616
75 641
A Table of Natural Sines.
65
75 028
880 251
24 751 158
24 394 57
56
23 679 55
74 822
74 615
74 408
888 943
73780
73569
51
50
48
47
46
915016
19 682
42
41
18 574
18 204
22761 390
38
17086
16 712 36
د د د د پي
a NO
16 337
26772991
15 961 34
15584
28 1778 791
DI 4 Deg. 85 Deg. 5 Deg. 84 Deg. 6 Deg. 83 Deg. 7 Deg. 82 Deg. M
01697.5:65 / 9975.641 877 55719961.947 1045.285 9945.219 1218.693 9925.462 60
I 700467
75 437
874 455
61 693 | 1048178
59
44 914 | 1221581!
25 107
2 703 368
75 233
877 353
61438 1051 070
44 609 1224 468
31 706 270
61183 1053 963 44 303 1227355
4:709 171
883 148 60 926 1056856 43 996 | 1235241 24 037
5 712073
886 046 60069 1059 748 43 688 | 1233 128
6
714 974
60 411 1062 641
54
43 379 | 1236015 23 319
7717876 74 199
891 840 60 1521005533
53
43070 1238 gor
22 959
8
720 777 73990 894 739 59 892 1068 425
52
42 760 1241788
22 599
9 723 678
897635 59 631 1071 318 42448 1244 674 22 237
10726 580
900 532 59 370 1074 210 42136 1247 560 21 874
II 729 481
73357
49
903 429 59 107 1077 102 41 823 1250 446 2I SII
12 732 382
73 145 ос6 326 58 844 | 1079 994 41 510 1253 332
2 I 147
13735 283 72231
909 223 58 580 1082 885
5
41 195 1256218 20 782
14 738 184 72 717 912 119 58 315 1085 777 40 880 1259 104
20 416
15 | 741 085 72. 502
58 049 1088 669
45
40 563 1261 990 20 049
16
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1434 926 9890 514 1607 426 9869 964 177943. 1340 407 | 1950 903 9807853 45
1437 805 98960901610 297 9369 4961782278 1839 889 1953 750 9807 285 44
1140 684 9895 677 1613 167 9309027 1785 150 1839 370 1956 609 980671643
1443 562 9895 256 16 16 033 5800 5571788 0222838 850 1959 401 9806 147
19 1446 440 98948381 1618 90.) 9808089 17908841883.30 | 1962 314 9505 576 41
14:49 319 9894 4161 1621779 9867615 1793 740 9937 008 1965 166 9805005
1452 197 19875 9:41 1624650 9867 143 1196 607 9837 285 | 19630139804 433 39
1455 075 9893572 1627 520 9866 670 1799 469 47836763 1970870 9863 86038
1457 953 9893 148 1630399 9806 176 1802 3300336 237 | 1973722 9303 286 317
1460 830 9892 723 163.3 260 9865722 1805 191 9835 715 1976573 9302 712 36
25 1463 708 9892 298 1636 129 9865 246 1308 052 9835 189 1979 425 9802 136 35
26
1466585 9891872 1630999 1 98647701310913 9834663 1982 2769801 560 / 34
27 1469 463 9891445 1641 868 9864293 1813774 19834136 1985 127 | 9800 983 33
28
1472 340 98910171644 738 9863815 1816 635 9833 608 1987973 9800405
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29
1475 217 9890588 1647607 9863 336 1819 495 2833079 1990824 2799 827 31
30 1478094 9890 159 16504769862 856 1822 355 19832 544 1993679 2799 247 30
1480971 9889728 1653 345 9862 375 182521598,3201) 19205309798667 39
32 14838489889297 1656 2,14 0861894 18280755331 487 1999 380 9798086 28
33
1486 724 9888 8651659 082 9861412 1830 935 9830955 2002 230 9797 50427
34 1489601 9888 432 166195119860929 1833 795 19830422 2005 080 9796921 26
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1492477 9887 998 1664 819 9860445 1836 654 1829 888 2007 930 2796337 | 25
36
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37 1498230 9887 128 | 1670 5569959475 1042 373 9828 818 2013 620 9795 167 23
1501 106 9886 692 1673 423 0358 9381845 232 4828 282 2010 478 9794501 : 22
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2608 9884 939 1684 894 9357035 1836 6669326 128 2027 373 9792 228 18
43 | 1515 484 9884 498 1689 761 9856544 1859 524182.5587 2030 721 9791638 17
1518 359 9284057 1090 628 6050053 1862 382 9825046 2033 569
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1524109 9883 172 1696 362 9855068 1868 098 9823961 2039 265 | 9789862 14
47 1526984 9882723 1699 2289854 574 1870 950 9823417 | 2042 113 9789 268 13
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53 1558598.9877 792 1730 752 9849086 | 1902 379 9817380 2073 426 0782 684
59 1561 472 9877 338 1733617 9848982 1905 234 9816 826 2076 272 2782080
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12 2113 248 9774 159 2283 509 9735 789 2453074 9694453 2621 892 9650 165 48
13 2116 091 9773 544 2286 341 9735 124 2455 874 9693 740 2624 699 9649 402 47
14 2118934 9772928 2289 172 9734 459 2458713 9093 025 2627 566 9648638 46
2121 777 9772 311 2292 004 9733 793 24615339692 309 2630 312 9647 873 45
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311 2167 236 9762 3302337 282 9723 020 2506 616 9680748 2675 187 963552729
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34 2175 754 9760 435 2345766 9720 976 25150639676 557 26335949633 189 26
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35 2178 5939759 002 2348 394 9720 294 12517879 9677825 2686 3969632 408 25
36 2181 432 9754 168 2351 421 9719610 2520694 9677092 | 2689 198 9631 626 24
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37 2184 271 9758533 2354 248 9918 926 2523508 9676358 2692 oco 9630843 23
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38 2187 110 9757897 2357 075 9718240 2526 3231 9675 624 2694 801 9630 060 22
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40 | 2192 7869756623 2362 729 9716 367 253.1952 9674 152 2700 403 9628 490 20
219.5 6241 9755985 2365 555 9716 1802534706 9673 415 2703 204 9627 704 19
42 2198 462 9755 3452368381 ( 9715 491 | 2537579 0672 670 2706 004 9625 917
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43 2201 300 9754 706 2371 2072714 802 2540 393 9671939 | 2708 805 0626 130 17
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45 2206974 2753 423 | 2376 859 0713421 2546019 19670459 2714 404 9624 552 | 15
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2215 4856751 494 2385 335 | 9711243255-4458 9668234 2922E02 9622 180
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02756.374 9612.617 2923.717 0563.048 3090.190: 9510.565 3255.6821 9455.180 60
I 2759 170 9611 815 | 2926 499 9562.197 301)2 936 i 9509 666 3258 432 2454 238 59
22761965. 9611012 2929 280 9561 345 3095 7021 9508766 3261 182 9453 250 | 58
3 2764 761 9610 208 2932061 : 9560 492 3098 468 9507865 3263 932 2452 341 57
|
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4 27675569609 403 29348422559639 3001 234 9506963 3266687 9451351 56
5 512770 352 9608 598 | 2937 623 9558 785 9103 999 95060613269 430 9450 441 55
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2773 147 9607 792 29404039557 930310676495051571 3272179 9449 489 54
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7 2775941 9606 984 2943 1839557074 3109 529 950,4253 3274 928 9448537 53
812778736 9606177 2945 963 9556 218 9503348 3277676 9447584 52
92781530 9605 368 2948 743 9555 361 3115 058 95024143 3280 424 9446 63051
3112 290
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10 2784 324 9604 558 2951522 9554 502 3117 822 9501 536 3283 172 9445 675 50
II 2787 118 9603 748 2954 302 9553 0-13 3120 586 9500629 3285919 9444 720 49
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12 2789911 9602937 2957081 9552 784 3123 349 9499 721 3288666
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132792 704 9602 125 2959859 9551 923 3126 112 9498812 3291 413 9442 807 47
|
14 2795 497 1 9601 312 2962638 9551 062 | 3128875 2497 902 3294 160 9441 84946
15 2798 290 9600 499 2965 416 9550 199 3131638 9496991 3206 906 9440890 | 45
16 2801 083 9599 684 2968 1949549 336 | 3134 400 9496 080 3299 653 9439 931 44
17 2003 875 9598 869 2970971 9548 473 3137 163 9495 168 3302 398 2438 971 43
18 2806 667 9598051 2973749 9547 608 3139 925 | 9494 255 3305 144 9438010 42
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19 2809 459 9597 236 29765269546 743 314 2 686 9493 341 3307 8899437 048 41
20 2812251 9596 418 2979 303 9545 876 3145 448 9492 426 3310 634 2436 085 40
21 2815042 9595 000 29820799545 009 3148 209 9491511 3313379 9435 122 | 39
22 2817833 9594 781 2984856 9544 141 31509691 2190595 3316 123 9434 157 38
2312820 624 9593 961 2987632 9543 273 31537301 9489 678 3318 867 9433 142 / 37
24 2823415 9593 140 2990408 9542 403 3156 490 94007603321611 9432 227 36
|
25 2826 205 9592 318 2993 184 9541 533 3159250 94878423324 355 2431 260 35
2612828995 9591 496 2995 959 9540 662 3162 010 94869223327098 | 9430 293 34
27 2831785 9590 672 2998734 9539 799 3164770 9486002 33298419429 324 33
|
28 2334 575 9589 848 3001 509 9538917 31675291 9485081 3332 584 9428 355 32
29 2837 364 9589 023 3004 284 9538044 3170288 9484 159 3335 326 9427 386 31
30 28401539588 197 3007 058 9537 170 3173 047 9483237 3338 069 9426 415 | 30
31 2842 942 9587 371 3009 832 9536 294 3175805 9482313 3340 810 9425 444 29
32 2845731 9586 543 3012 606 2535 418 3178 363 9481389 3343 552 9424471 28
33 2848 520 9585 715 3015 380 9534 542 3181 321 9480464 3346 293 9+2349827
3181321 |
34 2851 308 9584 866 3018153 0533 664 3184 0799479538 3349 034 9422 525 | 26
35 2854 096 9584056 3020 926 9532 786 3186 836 9478 612 3351 775 9421550 25
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3628568849583 226 30236929531907 3189 5931 9477684 3354516 9420575 24
37 2859671 9582 394 3026 471 9531 027 3192 350 94767563357256 9419598 | 23
38 2862 458 19581 562 3029 244 9530 146 3195 106 9475 3273359 996 10418 621 | 22
39 2865 246 9530 729 3032016 9529 264 3197 8539474897 3362 735 9417644 21
|
40 2868032 9579 895 3034 788 9528 382 3200 61019473 460 3365 475 9416005 20
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41 2870819 : 9579 060 3037 559 95274993203 374 6473035 3368 214 9415 686 19
42 28736059578 225 3040 331 9526 615 3206 130 0472 103 | 33709539414 70518
432876391.9577 389 3943 1029525 730 3208 8851947:17013373 691 9413724 17
44 2379 177 9576552 3045 872 9524 844 3211 6401 9470236 3376 429 9412 743
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45 2881963 9575 714 30486439523 958 3214 395 19469301 3.379 165 9411 76015
462884 748 95748753051413 9523071 32171499468 366 3361 905 941077714
47 2887533 9574 035 30541839522 183 | 3219 903 9467 430 3384 642 9409 793 13
48 2890 318 9573195 3056953 9521 294 32226579466 493 / 3387 379 12408 308 ! 12
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491 2893 103 9572 354 3059 723 9520 404 3225 411 9465 555 3330116 9.407822 | 11
502895 887 9571512 30624929519514 3228 164 9464616 3.592 852 9400 835 10
512898671 9570 6693065 261 3 30 917 9463677 3395 589 9405 818
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52 2901 455 : 9569 825 36680309517 731 3233 670 9462 736 3398 325 | 9404860
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3420.201 | 9396.926 3583.679 9335.804 3746.066 9271.839 3907.311 | 9205.049
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2 3425 668 9394 935 3589 110 9333 718 3751.459 9269 658 3912
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3 3428 400 9393 9383591 825 9332 673 3754 1569268 566 3915 343 9201 635 57
43431 133 9392 940 3594 540 9331 628 3756852 9267 474 3918 019 9200 496 56
5 3433 865 9391942 3597 254 9330 582 3759 547 9266 380 3920 695 9199 356 55
63436 597 | 9390 943 3599 968 9329 535 3762 243 9265 286 3923 371 9198 215 54
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50664
09198
38 741
36935
}
73 758
44 146
26
45 484
43 756
42 028
18 314
20 592
51 800
54046
28
M 36 Deg. 53 Deg. 31 Deg: 52 Deg. 38 Deg. si Deg. 39 Deg. 50 Deg. M
0 5877.8538090.170 6018.1507986.355 6156.615 7880.1086293.204 7771.460 60
I 5880 206
88 460 20473
84 604 1 6158 907 78316 6295 464 1 69 629 159
2 1 5882558
22 795
82 853 ; 6161 198 76524 6297 724 67 797:58
3 5884910
85037
25 117
81 100 6163 489 74 732 6299 983
65 965 : 57
4 5887 262
4
83 325
27 439 79 347 6165 780 72 939 6302 242 64 132:56
$ 5889 613
S
81 612
77 594 6168069
71 145
04 500
62 298:55
65891964 79899
75 839 6170359
06758 60 464 i 54
75894 314 78185 34 400 74 084) 6172648
09 OIS
58629 153
8 5896663 70470
36 719
72 329 6174936
II 272
56 794 52
95899 012
74 754
70 572 6177 224
63 963
13528
54957 51
IO 5901 361 73038 41 356
68 815 6179 511
15 784
53 I2I : 50
il
03 709 71 321 43 074
6 67 058 6181 798
51283:49
12
69603 45 991 65 299 6184084
58 569 20 293
49 445 : 48
13
67885 48 308 63 540 6186 370
22 547
47 606 47
i
14
10 750
66 166
61 780 6188655 54970
45 767 46
15 13 096 64 446 52 940
60 020 6190939
53 169
27 053 43 926'45
16 I5 442
62 726
55 255 58 259 6193 224
51 368
29 306
42086 44
17 17787
57 570 56 497 6195 507
31 557 40 244 43
18
20 132
54 7356197 790
47 764
38 402:42
19 22 476
52972 6200073 45 961 36059
36559 : 41
24819 55837
64 511
44 157 38 310 34 716:40
54113
66 824
49 444 04 636 42 352
40559 3287239
29 505 52 389 69 136
40577
31 027 38
23
71 447 45 913
45 037
29182 37
24 34 189
48938
II 478
47 305 27 336 1 36
:25
36 530
47211 76069
42 379 13 757 35 127 49 553
25 489 135
38 871
78 379
40 611
16 o36 33 320
23642 134
27 41 211
80 689
38 843
31 5II
21 794 33
43550
82998
37074
29 702 56 292 19 945 32
29
40 299
85 306 35 304 22 870 27892 58537 18096 31
48228
38 569
87614
33 533
26082 60 782
31 50566
36 838
89922
27 423 24 270
14 395 29
52 904 35 107 92 229 29 990 29 698
65 270
12 544 28
55 241 33 375 94 535 28 218
31974 20646
10 692 27
34
57 577 31 642
18833
69 756 08 840 26
35 59 913 29 909 6199 147
17019
71998
OI 452
22 896
38 796 15 205
74 240 05 132 24
64 584
21 I21 41069
13390
76 481
03 278 23
66918 24 705
06 060
19345 43 342
78721 OI 423 27
39 69 252
22 969
08 363 17 569
09 757 80961 7699 567 21
42
I5 792
07 940 83 201 97 710 20
73919
19 495
12969
14 014 50156
95853 19
42
76251
17 756
IS 270
12 235
52427
87 678 93 996 18
43 78583 16 018
17572 IO 456
54696
02 485
89916 92 137 17
44
80915
14278
08 676 56966 00 665 92153 90278 16
45 83 2:46
12 5,38 22 173
06896
59235 7798 845
94 390
IS
85577
24473 05 115 61503 | 7797024 96 626
47 87 906 09056
26772
03 333 63 771 17795 202 98862
48
07 314
OI 550 66038 7793380 6401 097
05 571
31 3697899 767 68 205 779155764033,32
50
03 827 33666 7897 983 70571 17789 7.33 6405 566
79 110 10
51
02 083 35 964 7896198 72 837 177879096407799
77 246
9
52 99 549
00338
39 260 17894 413 75 102 179860846410032 75 3828
53
OP 876 7998593
40556789262; 773667784 258 6412264 73 517
:54 04202 7996 847 428525890 841 796317782 4.316414 496 71652
6
:55
06528 7995 100
4.5 1477889054 81 894 7780 604 | 6416 728 69 785
5
:56
08854 799.3 352
47 4427889 266
8415777787771 6418958
67 918
4
57 III79 7991 604 49 736 7885 477 86420 77769496421189
3
13.503 7989855
52029 7883 688
88 682-7775 1201 6423 418 64 183
59
15827 7988 105 54 322 7881 898 SO943 177732906425 647
60 18150 7986 355
5661517580108 93204 777146016427876
H
45889
30
25146
16 246 1 30
31 762
63 026
32
22 459
-33
67 513
96841
26 445
34 248
24671
36522
06986 25
62 249
36
37
38
28 175
26.440
03756
II 574
71586
21 232
10666
45 614
47 885
41
06123
85 440
04 304
19 873
I
10797
46
88418
86 558 114
84 697 113
82 835 12
80 973 II
29071
49
I
90 236
92 565
94893
97 221
IO
Oo non per man
66 051
58
2
6
62 314
60 444
1
.
7年
​A Table of Natural Sines.
>
58 574
2
25 606
390II
SI 087
71560
04 266
06 424
43 461
45685
45 465
31 808
82516
+ 1 11
39 838
86 895
17206
19 361
14
61 240
02 181
93 458
95 645
97831
23668
25 821
00 225
53948
24 802
10721
60 300
22919
04 386
06570
08 754
64 532
66 647
17 268
88475
86515
84553
82592
81 199
!
17 482
97 262
75 IOI
61 748 34
19 662
49466
51612
M 40 Deg. 49 Deg. 41 Deg. 48 Deg. 42 Deg. 47 Deg. 43 Deg. 46 Deg. IM
.
06427.876 ; 7660.444 6560 5907547.096 6691.306 7431.448 6819.984 ; 7313.537
,
60
I
30 104
62 785
45 187 6693 468 29 502 22 III 7311 553 59
32 332
56 704
64 980
43 2786695 628
27554 24 2377309 568 158
3 34 559
54832
67 174
41 368 6697 789
26 363 7307 583 57
4 36785
52960
69 367
39 457 6699 948
23658
28 489 7305 597 56
5
375466702108
21708 30 613 730361055
6
41 236
49 214 73 752 35 634
19758 32 738 7301 623 54
7
47 340 75 944 33 721
17808
34 861 | 7299 635 53
78135
08 582
15857
36 984 97 646 52
9 47 909 43 590
80 326
29894
IO 739 13905
39 107 95 657 51
IO SOT32 41 714
27980 12 895 II 953 41 229 93 668 50
IT
52 355
84 706
26065
15 051
10 000 43 350 91 677 149
I 2
54 577 37960
24 149
08046 45 471 89 686 48
13 56 798
36 082
89 083
22 233
06092 47 591 87 695 47
59019 34 204 91271 20 316 2I515 04137 49 711 85 703 46
IS
32 325
18 398
51830
83 710 45
16 63 460
30 445
16 480
81 716 44
17 65 679 28 564
14.561 27 973 | 7398268 56 066
79 722 43
18 67 898 26 683 6600 017 12 641 30 125
96311
58184 77 728 42
19
70116
02 202
32 276
94353
75 732 41
20 72 334
08 800
34427 92 394
62 416
73 736 40
21 745ŞI 21 036
06879 36577 90435
71 740 39
22 76 767
19 152
04957 38727
69 743 38
23 7,8984
10 936 03034 40876
68 761 67 745 37
204
15 383 13119
OI III
43 024
70875 65 747 36
25
83414 13 497
15300 7499 187 45 172
72 988
63 748 35
26 85 628 II 611
47 319
80629
27 87842 09 724
95 337
78666 77 213 59 748 133
28
90056 07 837 21 842
93 411
76703
79 325 57747 132
29
05 949
24022
53 757 74738
55 746 31.
30
26 200 89 557
55 902 72 773 83540 53 744 30
31
26692 02 170
87 629 58046 70808 85 655 51 741 29
32
98903
30557
85 701
68 842
87705
49 738 28
33 6501 114 7598 389 32 734
66 875
89 873 47 734 27
34 03 324 96498 34 910
81 842 64476
64.908
91981
45 729 26
35 05533
79 912
66 618
43 724 25
36 07 742 92 713
68760
41 719 24
37 09 951
41 437
70 901 59002 9@ 302 39 712 23
12 156 88 926
74117 73041
57032 | 6900407 37 705 224
39 14 366
87031
02 512 35 698 21
40 16 572 85136 47959 70251 77320
53 090 04617 33 69020
41
83 240
50131
79459
31 681 19
42
20 984
52 304
o: 824
2967118
43 23 189
54 475 64 446
83 734
47 173
I0927
27661 17
44 25 324
77548 56646
5
85871 45 199 I3029
25651 16
45 27 598
58817
88007
43 225 IS IZI 23 64,0 115
46
73 751
60987
58636 90143 41 250 17 232
21 628 14
47 32 004 71851 63156 56 699
92278
39 275 19 332 19615 13
69 951 65 325
94 413
37 299
21 432 17602 | 12
49
36 4053
96547
35 322 23 531 15589 1
38 609
66 148
so
33 345
13574
ΤΟ
64246
48941 6800813
31367
9
73994
02 946
09 544
8
53
45 209
45058
27 409
31922 07528
7
47 408
58535 78 326 43 115 07 209 25 429
OS SII
6
55 1
56 630
41 173 09339 23 449
36 114
03 494
56: SI 804 54 724.
82655 39229 II 469 21 467
·38 209
01 476 4
54 002
84 818
37 285
13599
40304 7199 457 3
56 198 50 911
86981
35 340 15728
17503 42 398 7197 438
591 58 395
49 004
89 144
33 394 17856
15 521 44491 7195 418
601 60 590
47 0961 91306 31448 19 984 13537
465847193 398
92268
94480
91 484
81 435
04060
28 379
00 280
60190
62 333
83 772
94606
37087
62940
60971
94089
Nau
39 262
77981
96195
90 820
76 049
38
43 612
45785
72 184
75181
S5 061
18778
06721
68 317
66 382
81 343
81597
51118
49146
79446
62 510
60 574
75 650
29 801
48
34 206
68050
67 493
54 760
52321
II
881
98681
so
51
52
69 661
71 828
25630
27 728
29 825
40810
43 CIO
II 959
62 343
60 439
4699?
29 388
76160
05 078
54
34 018
49 607
80 490
cah in a
O am
Common av
52 818
19 486
57
58
2
A Table of Natural Sincs.
75
5
M 44 Deg. 45 Deg. M
50767
52 858
60
O 6946.584 7193.398
I
48 676
2.
89 355
3
41 54 949
51.57 039
81 263 154
7
61 217
8 63 305
9
65 39%
67 479
69 565
91 377 59
58
87 333 57
65 310 56
83 287 15
59 128
...
65 049
46
82 071
1884 153
*19
86 234
88315
20
2/
22
1
98.711
79 238 53
77 213 52
75 187 51
IO
73161 50
II
71 134 149
I2 71 651
69 106 48
13
73 736
67 078 47
14
75 821
15 77 905
16
63 019 45
79 988 60989 44
IZ
58959 43
56927
42
54 895 41
52 863 40
90°396
5083039
92.476
48 796/38
23 1 94 555 46 762 37
24
i 96 633
44 727 / 36
25
26170001789
: 42 бух | 35
40055 34
27
02'866
38618 33
36581/32
29
34 543 35
31
30 09093
32 504 30
31
II 167
30465 129
32 13241
28 426 28
33
IS 314 26 385 27
34
17 387
35
19 459
3.
20 260 24
37
18218 23
38
3
27 741 14 130
12086 20
41
31 879 10 041 19
33 947
18
OS 948 | 17
05
38081
16
40 147 01 854 15
42 213 7099 806 14
28 : 04 942
1
07 018
د د د د در
24 344
22 303
26
25
1
21 531
23601
25672
22
16 174
24
39
20
29 817
IO
07995
42
!
43
36014
03 901
..
| 45
$0
47
48
50
52
53
54
55
44 278 7097 757
13
I
46 342 7095 707
II
49 48 406 7093 637
50 469 | 7091 607 10
9
51 52 532 7089 556
8
545947087 504
5665:
57085 451
7
6
58716 7083398
60 7767081 345
S
62 835 7079 291
4
64 894 7077 230
3
66953 7075 180
69 011 7073114
-71 0681 7071.008
CON ah p co
50
57
2
58
I
mer
3
1
*
{
4
+
1
1
។
1
1
4
30*
SSCH
1
T
B
L
E
ii
0 F
1
LOGARITHMIC VERSED SINES
1
!
f
3
2
To every Minute of the Quadrant,
1
1
26
1
I
!!
ANI,
'w
ܝܟ ܂
erwerp
{
t
A Table of Logarithmic Verſed Sincs.
77
M
M
o Deg.: i Deg.
2 Deg.
3 Deg.
4 Deg.
5.Deg. M
0
I
I
2
2
II
4494578
I
6.1827137 6.7847406, 7.1360000 7.3866083 | 7.5803891
2.62642221 1970705
7919481
1416791
3902785 5832778
23 2284822
2111938 7990963
1464636
3939736 5861568
3 35806647 2250912
8061861
I512219 3974539 5890263
3
4 38305422 2387696 8132185 1559542 4010196 5918864
4
S 40243622 2522360
8201944
1606609
4045706
5947370 5
641827246 2654968 8271147 1653422 4081071
5975783
6
743166182
2785581 8339803 1699984 4116293 6004103 7
84 43.26020 2914259
.
8407920 -1746297 4151372 6032331
8
9 4 5349070 3041058
8475507 1792365 4186311 6060468 9
10 4 6264219 3166033
8542572
1838189 4221109
6088513
IO
4 7092072 3289235 8609123 1883773
4255767 6116468 | 11
12 | 47847843 3410714
8675167 1929118 4290288 6144333 12
1348543084
3530516
8740714
1974228 4324673 617210913
14 49186777 3648089 8805768 2019104 4358921 6199796 14
1549786041 3765275 8870340 2063750 4393035
6227395 | 15
165 0346614 3880317 8934434 2108167 4427015 6254906 16
17 1 50873192 3993855
8998059
2152358 4400862
628233017
1851369663 4105928 9061221 2196326
6309668 | 18
19 5 1839283 4216573 9123927 2240071 4528163
6336920 19
2015 2284810 4325826
91861831 2283597 4561619 636408620
21 52708595 4433722 9247996 2326906 4594946 639116721
2253112661
4540294 9309372 2370000
4628146 641816422
23 5 3498763
4645573 9370317 2412881 4061219 6445078 23
24 5 3868430 4749592 9430837 245 5551
4694166 6471908 24
25 54223003 4852380 9490939 2498013 4726989 6498655 25
26 5 4563669
4953965
9550627
2540267 4759688 6525320 26
275 4891475 5054376 9609907 2582317
4792264 6551903 27
285 5207359
5153639 9668786 2624164 4824719
657840428
29 15 5512157
5251780
9727268 2665810 4837052 6004825 29
3055806620 5348825 9785359 2707258 4889265 6631166 30
315 6091427 5444797 9843063 2748508 4921359 6057427 31
3256367191 5539720 9900387 2789563 4953335 6683008 32
33 56634468 5633616 9957334 2830425 4985193 670971133
34 S 6893765 5726509 7.0013911 2871095 5016934 6735735 34
35 57145546 5818418 0070121 2911576 5048560
6761682 35
365 7390233
5909365 0125969 2951869 5080071 678755036
5 7628215 5999369 0181461
2991975
5111468
6813342 37
38 57859950 6088450 0236600 3031897 5142751 6839058 38
39 58085468
6176626 0291391 3071636 5173923
6864697 | 39
40 58305373 6263916 0345838 3111194
5204982 689026040
41585 19848 6350337 0399946 3150572 5235931 | 6915749 41
425 8729154
64359071 0453719 3189773 5266769 6941162 42
435 8933535 6520642 0507161 3228797 5297498 696650243
44 59133217 6604558 0560276 3267646 5328119 0991767 44
4559328411 66876711 0613068
3306322 5358632 7016959 45
|
46159519314 6769996 0665540 3344827 5389038 704207846
4759706112 68515471 0717698 3383161 5419338
7067124 47
485 9888977 6932340: 0769544 3421327 5449532 709209848
49 60068070 7012389 0821082 3459326 5479621 711700149
50 60243546 7091706 0872316 3497159 5509607
714183250
5160415546
7170305 0923249 3534828 5539489
7160592 51
5260584206 7248199 0973885 3572334 5569268
719128152
5360749654 7325400 1024228 3609678 5598946 7215900 53
54 60912008
7401921: 1074280 3646863 5628522 724045054
S5 6 1071384 7477774 II24045 3683888 5657998
726493055
506 1227887 7552970 1173527 3720757 5687373 7289341 56
$761381620 7627520 1222728 3757469 5716650 731368357
5861532679 7701436 1271652 3794027 5745828 733795858
5961681156 7774728 1320302 3830431
5774908
7362164 59
60 ! 6 18271371 7847406 : 1368680 3866683 $8038911 7386303 1 60
37
78 A Table of Logarithmic Verſed Sines.
M 6 Deg 7 Deg.
8 Deg.
9 Deg. 10 Deg. II Deg. M
.
O
I
2
I
2
@ we
2746082
IZ
20
21
22
29
0 17.7386303 7.6723806 7.9881990 18.0903166 8.1816220 18.2641757
7410375 8744436 | 7 9900038 0919203 1830648 2654867 1
7434380
8705017 79918047 0935210 1845051 2667957
3 7458319 8785550 79936020
0951188
1859431 2681028
3
7482192 88060,33 79953955
0907136 1873786 269407814
5 7505999 8826469 79971853 0983055 1888118
2707109! S
7529742 884685679989713 0998944
1902426
2720119! 6
7 7553419 8867196 | 8.00C7537 IOI4804 1916710
2733111
7
8
7577031
8887487 0025325 Io30635 1930971
9 7000580
8907732
0043076
1046437
1945208
2759935
9
Іо
7024064 8927928 0060790 1062211
1959421 2771967, 10
II 7647485 8948078 0078468 1077955
1973611 2784880 II
7670843 8968181 0096110 1093671 1987778
2797774 12
13 7694138 8988238 0113716 1109358 2001921 2810649 13
14 7717371 9008248 0131287 1125017 2016042 2823504 14
15 7740341 9028212 0148822 1140647 2030139 2836341 15
16 7763649 9048130 0166321 1156249 2044213 2849158 16
17 77866901 9068002 0183785 1171823 2058264 2861956 17
18 7809682 9087829 0201213 1187369 2072293 2874735 18
19 7832607 2107610 0218607 I202887 2086298 2887495 19
7855472 9127346 0235965 1218377 2100281
2900236 20
7878276 9147038 0253289 1233840 2114241 2912958 21
7901020 2166684 0270578 I 249274 2128179 2925661 22
23 7923705 9186286
0287833 1264681 2142094 2938346 23
24 7940331 9205844 0305053
1280061 2155987 2951012 24
25 7968897 9225358
0322239 1295413 2169857 296366025
26
7991405 9244827 0339391 1310738
2183705
2976289 26
27
8013855 9264253 0350508 1326036 2197531 2988899 27
28 8036246 9283636 0373592 1341307 2211334
3001491 28
8058580
9302975
0390643
1356551 2225116
3014064 29
30
8080856 9322271 0407659
1371768
2238875 3020619 30
31
8103075 9341523 0424642 1386958 2252613 303915631
32
8125237
9360734 0441592 I 402121
3051675 32
8147343
9379901 C458509 1417258
2280023 3064175 33
8169392
9399027 0495393
1432368 2293695
307665734
8191386 9418110 0492243 1447452 2307345 308912235
8213323
9437151 0509061 1402510 2320974 3101568 36
8235205 9456150 0525846 1477541
2334581
3113996 37
8257032 9475107 0542599 1492540 2348167 3126406 38
8278804 9494023 0559319 1507525 2361732 3138798 39
8300522 9512898 0576007 1522478
2375275 315117240
8322185 9531732 0592663 1537405
2388797 316352941
8343794
9550525 0609286 1552307 2402297 3175868 42
8365349 95692761 0625878 1567182 2415777 3188189 43
44
8386851 9587988 0642438 1582032 2429235 3200493 44
8408299 9606659 0658966 1596857 2442673 3212779 | 45
8429695 1 9625290 0675463 1611656
2456089 3225047 | 46
84510371 9643880 0691928 1026430 2409485 3237298 47
48
8472327 9662431 0708362
1641178 2482860
324953248
8493565 9680942 0724764 1055902 2496214 3261748 49
50 8514758 9699414 0741136 1670600 2509547 3273947 50
8535885 9717846 0757470 1685273
2266329
33
34
35
36
37
38
39
40
41
42
43
45
46
47
49
2522860
328612851
8556968 9736239 0773786
1699921 25361521 3298292 52
8577999 9754593 0700065 1714545 2549424: 331043953
85989801 9772908 08c6313 1729144 2562675 3322569 54
8619910 9791184 c822531
1743718 2575906 3334682 55
8640749 9809422 0838718 1758267 2589117' 334677856
57
8661618 0827021 0854875 1772792 2602307 335885757
$8
9845782
0871002
1787292 2615477 3370918 58
59 8703126 9863905 0887099
1801768 2628627 3382963 59
60 8723806' 98819901 0903166 18162201 2641757
3394991 160
SI
52
93
$4
55
56
8682397
A Table of Logarithmic Verſed Sines. 79
M, 12 Deg. 13 Deg. 14 Deg. 15 Deg. 16 Deg. 17 Deg. M
1
I
2
3454880
6454909
II
IS
4317216
م تي
of 8 3394991 / 8.4087475 8.4728189 8 53242531 8.5881406 8.6404342
3407002 4098556 4738472 5333844 5890390
6412791
3418997 4109622 4748742 5343423 5899365 6421231 2
3 3430975 4120675 4759000 5353992 5908330 6429663 3
4 3442936 4131713 4769246 5362551 5917286 6478087 4
5
4142736
4779480
5372098 5926233 6446502s
6 2466808
4153746 4789701 5381635 5935170
7
7
3478719 4164741 4799910 5391161
$ 944097
6463308 6
8 3490614 4175723 4810107 5400677 5953016 | 6471698 8
9 3502492 4180690 4820291
5410182
5961925 6480080 9
IO 3514354 4197644 4830404
5419676
5970824 | 6488454 10
3526200 4208583 4840025
5429160
5979715 6496820 11
12 3538029 4219508 4850773
5438633 5988596
6505177 | 12
13
3549842 4230420 4860910 5448096 5997468
6513526 13
14 3561639 4241318 4071034
5457548
6006330 6521807 14
3573419 4252201
4881147 5966990 6015164 653020015
16 3585184 4263092 4891246 5470422 6024028 6538524 16
17 3506932 4273928 4901336 5485843 6032863 6546841 17
18 3608664 4284770 4921412 549,5253 6041684 05551918
19
3620381
4295599 497.1477 55046541 60505c6 6563449 19
20 3632081 4306414 4931530 5514044 6059313 657174120
21
3643765
4941572 5523423
6068112
658002521
22 3655434
4328004 4951601 5532793 6076901 6583301 | 22
23
3667086 4338778 4961619 5542152 6085681 6596569 / 23
24
3678723 4349539 4971625 5551500 6094453 6604829 | 24
25 3690344
4350286
4981619 5,560839 6103215 6613081 25
26 3701950
4371020
4991001 5570167 6011968/ 0621324 25
27 3713539 4381740 5001573 5579485 6120752 662956027
28
3725114 4392447 5011532 5588703 6129448 6637788 28
29
3736672
4403141 5021480 5598041 6138174 6646008 29
30 3748215
5031416 5607379 6146891 6454220 30
31 3759743 4424488
5041341 5616650 6155600 6662424 31
32
3771255 4435142 SO51254 5625924 6164299 667062032
33
3782751 4445783 SO61156 5635181 6172990 607&808 33
34 3794232 44.56410 5071046 5644429 6181672 6686988 34
35 3805698 4467024 5080925 5653666
6190345 6695100 35
3817149
4477625 5090792 5662894 61990091 6703324 | 36
37
3828584 4488213 5100048 5672111 6207664 (711481 37
3840004 44.98788 SI10493 5681318
6216311 6719630 38
39
3851499 4509350 S520326 5090516
Wao
4413821
36
38
6224948 672777139
40
3862799
4519898 5130148 5699704
6735904 40
41
3874174 4530434 5139959
5708881
6242197
6744029 41
42
3885533 4540957 5149758 5718049
6250809 6753147 | 42
43
3896878 4551467
5159540 5727207 6259412 070025643
44
3908207 4561964 5169324 5736355 6268000 676835844
45 3919522 4572448 5179089 5745494
67,6453 45
46 3930822 4582920 5188844
5734622
6285168 6784539 46
47
3942107 4593378 5198588 5703741 6293736 6792613 | 47
48
3953377 460382004 5208320 5772850
6301205 očoc08948
49 3964632 4614257 521802 5781950 6710046
50 3975873 4624677 5227752 5791039 6319388! 6810409 50
51 3987098 4635085 5237451 5900119 6327922 1 6224057 SI
52 3998310 4045480 5247140 5809189 6336447 663289752
53
4009506
4655863 5250817 5918250 634446 684043053
54
4666233 526684 5027301
6353472 6348956 54
55 4031855 4076590 5276139 5636342
6361971
56 4043008
: 4680435
5285784 5825.74 6370402 6804984 56
57 405 4147 4007267 $295417 $854396 6376945 687298657
58 4065270 4707587 5305079 5863409 6387419 688048158
59 4076380 4717894
587 412 6395681 688896y59
60 4087475 4728189 5324253: 5881106
5381.jpg
6896949160
6233577
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68c6753 49
بده
4020688
(&$6973 55
652ܝ1ܪܢ 5
6404342
80 A Table of Logarithmic Verſed Sines.
O
I
I
2.
2
OOOOert w O
!
8263759
7422564
10
II
IE
14
20
22
24
M 18 Deg., 19 Deg, 20 Deg. 21 Deg. 22 Deg. 23 Deg: M
018.0&96949 8.7362485 8.7803705 8,8222901 8.8622277 18.9003406
|
6904921
7370030 7830866
8229774 8628774 9009613
6912886
7377570 7818022 8236582 8633265 9015816
3
6920844 7335102 7825171 8243385
8641752
9022013
3
4
6y=8794
7392628 7832314 8250182 8648233 9028207
4
5
oy36736 7400147 7839452 8256973 8654710
|
9034395 5
6 6944672 7407659 7846583
8661181
90405 79
6
6952599
7415165 7853708
8270539
8667648 2046759 7
8
6900520
7860827 8277314
8674109 9052934
8
9
6908232
7430156 78679101 8284083 8680566
9059104
9
6976338 7437642
7873047 8290848 8687018 906527010
0984236 744.5121 7882149 8297606 8693464 9071431 | LE
I2 6992127 7452593 7889244 8304360 8699906 9077588 12
13 7000010 7460059 7896333 8311107 8706342 9083.940 13
7007886 7467518 7903416 8317850
8712774
9089887 14
15
7015755 7474971 7910494 8324587 8719201 909603015
16 7023617 7482417 7917565 8331318 8725623
9102169 16
17 7C31471 7489857 7924630 8338044 8732040 910830317
18
7039318 7497290 7931690 8344765 8738452 9114432 | 18
19
7047158 7504716 7938743 8351480 8744859 612055719
7054990
7512136 7945721 8358190 | 8751201 912667820
21 7062815 7519.549 7952833
8364895 8757658
913279421
7070633 7526950 7959869 8371594 8764051 9138905 22
23
7078444 7334357 7966899 8378288 8770438 9145012 | 23
7086247
7541751 7973923 8384976 8776821 915IIIS 24
25 7094044
7549138
7980941 8391660 8783198 9157213 25
26
7101833
1556519 7987953 8398337
8769571
916330626
27
7109615 7503894 7994960 8405010
9169396 27
28
7117390
7571262 8001961 8411677
8802303
917548028
29 7125157 7578623 80089568418339 8808661 918156129
30 7132918 7585979 8015945 | 8424996 8815014 9187636 30
31 7140071
7593327 8022926 1 8431647 8821363 919370831
32
7148418 7600670 P029906; 8438294 8827707
9199775 32
33
7156157 7603006
8.36877 8444934 8834046 9205 837 33
|
34
7163839 7615336
8043843
8451570
8640380 921189534
35
7171014 7622659 8050803 84582_o 8846710 9217949 35
7179332 7629976 8057758 8464826
8853034
9223999 36
37
7187044
7037286
8064707 8471445
8859354
9230043 / 37
38 2194748 7644591 80716498478000 8805069 923608438
39
7202445 7655889
8078587 8484670 8271980 9242120 39
40
7210135
7659186
8085518 8491274
8795939
35
بی بی سی بی
I
8878285 9248152 40
41
7217818 7606466 80924441 8497873 8884586 925417941
42
7225494
7673745 8092364 ; 8504467 8890882 926020242
43
7233163 7081018
8106278, 8511055 8897173 9266221 43
44
7240825 76%8284
8113187 8517639 8903400 9272235 44
45
7248480 7695544 8120090 8524217 8909742 9273245 | 45
46 7256129
7702718
8126988 8530790 8916019 9284251 46
47
7263770
7710046
8133879 8537358 8922291 929025247
48 7271404
7717288
8140765 8543921 8928559
924624,948
49
7279032 7724523
81476461 8550479 8934822 9302242 49
7286653 7731752
8154521
8557032 8941080 9308231 50
7294207 7738975 8101390 8563579 8947334 931421551
7301874
52
7746192 8168253! 857072 I 8953583 932019+52
53
7309474 7753403 8175111 8576659 8959827
54 7317067 7760607 8181964 | 8563191 8966066 933214154
7324054 7767805
8188810
55
8589718 8972301 9538108 55
56 7.332233 7774997 8195652 8596240 8778532
234407056
7339806
57
7782183 8202487 8602757 8984757 2350029 57
:58
7347373 7789203 82093171 8609268 Eg60978 9355983 58
7579.32 7 7,50537
59
82161421 8615775 8997194 93619,33 52
60 7362485 7803705 8222961. 8622277 9003400 9367878160
50
SI
170,53 هو
A Table of Logarithmic Verſed' Sines.
81
1
1
2
2
CON ah a W N O
4
9391618
II
13
9444785
نمک
OI60179
23
9526320
M 24 Dég. 25 Deg. 26 Deg. 27 Deg. 28 Deg.
29 Deg. M
089367878/8:97170352.00520619.0374005 9.0683803 9.0932293
|
O
9373819
9727731 0057531
0678767
0379265
0987176 I
9379756
972 3424 0062997 0384522 0693931
0992057
3
9385689
2734113
0068460
coyoyyo
0389776
09969347 3
9739797 C073920 0395026 0704040 ICCI8094 Ą
5
9397542 9745478
0072375 0.100273
0701049 10060,81 5
6
9403462
9751855
0084827 0105517
071.140
6
1011549
9409378 9756828
0090276 0110757
0719195 10164157 .
9415290 9762492 0095 721
0415994
072-1238 1021278
8
9
9421197 9768163 OIOI162
0.12 1 228
0724279 10261389
IO
9773824
9727101
OIC6600 0426-458 0734310
1030925 10
9433000 9779432
OII 2034 0431635 0739350
1035850 11
I 2
9436895
9785135 0117408 0436908 074+j1
104070112
9790785
0122092 0442122
0749409 104555013
14 9450672 9796431 0123315 04473-15
0;54434
1050395 14
IS
9450554 9802073 0133735 0-452559
0759455 105523815
16 9462433 9807711 0139151 0457769
0704474
106007816
17
$ 9468307 9813346 0144564 0162976 0769490 1064915! 17
18
-9474177 9810976 0149973 04081.o 0774502 1069749 18
12 94800.12 9824603 0155378 0.173300 0779511
10745EO 19
&
20 9485904 9830226 0160781 0478578
07643 i 8
1079408 20
21
9491761
98.35845
0483771 0709521 1081-37 21
22 9497015 9641460 017157-1 0.439902 0794521 1089056, 22
9503464 9847072 0176965 0.94119
0799513 109387023
24
9504304 9852679 0182353 0499333
OÓU.1512 1098693 24
25
9858283
:9515150
0187738
0809503
0504514
I103507 | 25
26
9520987 9813883 0193112 0509691
0817191
1108318) 26
27
9869480 0198496 0514865 0814470 111312627
28
9532648
9875072 0203670 0520036
0824458
I11793228
29 9536173
8 9880661 0209240 0525204
0829437. II22735 29
30 9544294 9886246 0214607 0530368
06 34413
1127534 | 30
31
95j0rio
9891827 0219970
0839380
0535523
I1;233131
32 2555922
9097404 0225330 0540687 0844356
113712632
33 9561731
9202973
0230087 0545842 0849322
I14I917 33
3+ 9507535 9908548 0236039 0550993
0834286 114070534
35
9573335 991411. 0241384 0556141 0059247 1151491 35
36
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37 9524923 9925235 025207? 0566422
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9656051 9947432 0273412 0536902
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0592008 009.087
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90577591 9258508
0284059 0597210
08,8824 110967543
44 9625355 9,264041 0289378 обс23 29 0903753
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9048370 9936134
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Co to co
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82.
A Table of Logarithmic Verſed Sines.
O
I
I
2
2
3
2345518
IO
II
1
22
M
M 30 Deg. 31 Deg. 32 Deg. 33 Deg. 34 Deg. 35 Deg M
9.1270225 9.1548276 9,1817061 9.2077136 9.2329007 9.2573136
1274938 1552831 1821466 2081400 2333139 12577142
1279649 1557382 1825868 2085661 2337267 2581145
3
1284356 1561931 1830268 2089920 2341393
2585147
4 1289062 1566477 1834065 2094177
2589147 4
5
S 1293764 1571021 1839060 2098432
2349640 2593144 5
1296464 1575562
1843452
2102684
2353761 259714016
2 1303161 1580101 1847842 2106934 2357879
2601133
8
1907855 1584637 1852230
2ΙΙΙΙ82
2361995 26051258
9 6312547 1589171 1856615 2115428 2366109 26091141 9
1317235 1593702 1860998 2119671 2370221
2613102 10
1321921 1598230 1865378 2123912 2374330
2617087 II
12
1326605 1602756 1869756
2128151
2378438 262107112
13 1331286 1607280
1874132
2132388 2382543
2625052 13
14 1335964 1611800 1878505 2136622 2386647 2629032 14
IS
1340639 1616319 1882876 2140854
2390748
2633009 15
16
1345311
1620835 1887245
2145084 2394847 2636085 10
I7
1349981 1625348 1891611 2149311 2398944
2640958117
18
1354648 1629859 1895974 2153537 2403038 2644929 18
19 1359313 1634367 1900336 2157760
2407131
2648899 19
20
1363975 1 1638873 1904695 2161981 2411222
265286620
21
1368634 1643376 1909051 2166199
2415310
2656832 21
1373290 1647876
1913406 2170416 2419306
2660795 22 -
23
1377944
1652374 1917758 2174.630 2423481
266475723
24 1382595 1656870
1922107 2178842 2427563
2668716 24
25 1387244
1661363
1926454 2183052 2431643
2672674 25
26
1391889
1665854
1930799
2187259
2435721
2676629 26
27 1396532 1670342
1935142 2191464
2439797
2680583127
1401173 1674828 1939482
2195668
2443871
2684534 28
29 1405811
1679311
1943819 2199868 2447942
30 1410446 1683791 1948155 2204067
2452012
269243130
31 1415078 5688269
3208263 2456079
2696377/31
1419708 1692745 1956819 2212458 24601.25 270032132
33
14.24335 1697218 1961147 2216650 2464208
270426233
34 1428960 1701689 1965473 2220839 2468269
2710202 34
35 1433581
1700157 1969797 2225027 2472328
271214035
36 1438201 1710623
1974118 2229212 2476385 2716075 36
37
1442817 1715086 1970437 2233396 2480440 372000937
1147431 1719547 1982754 2237577 2484493 2723941 38
39 1452042 1724005 1987068 2241755 2488544 2727871 39
40
1456651 1728461
2245932 2492593
273179940
1461257 1732914 1995090
2250106 2490640 273572541
1465861 1737365 1999997 2254279 2500684 273964942
43 1470461 1741813 2004302 2258449 2504727
274357143
44 1475060 1746259 2000605 2202617 2508767 2747491 44
45 1479655 I 750703 2012906 2266782 2512806 275140945
1434248 1755144 2017204 2270946 2516842 2755325 46
1488833 1759582 2021499 2275107
2520876 275923947
43
1493426 1764018 2025793 2279266 2524909
2763151 48
49 1498011 1768452
2030084
2283423 2528939
276706249
I 502594
1772883 2034373
2287578
2532967 277097050
SI 1507174 1777312 2038660 2291731 2536993
2774876 51
52
IS11751 1781738 2042944
2541017
277878152
53 1516326 1786162 2047226 2300029 2545039
278268353
54
1520898 179058 2051506 2304175 2549059
2786584154
55 1525467 1795003 2055783 2308319 2553077
279048355
56
1530034 1799419 2060058 2312461 2557093
279433056
57
1534599 1803833 2064331 2316601
2561107 279827457
1539161
1808245
2068602 2320738 2565119 280216758
591 1543720 1812655 2072870 2324874 2569128
28060581 59
60: 1548276 1817061 2077136 2329007 2573136 2809947160
28
268848429
1952488
32
38
1991320
41
42
46
47
so
2295881
58
A Table of Logarithmic Verſed Sines.
83
0
I
I
2
2
3051148
4
IO
II
12
>
2883475
20
21
22
M 36 Deg. 37 Dcg. 38 Deg. 39 Deg. 40 Deg. 41 Deg. M
019.2809747 9.3039829 (9.3263138 9.3480205 19:3691334 19.3896806
2813834
3043604 3266806
3483772
3694804 3900184
2817720
3047376
3270473 3487337
3698272 3903561
3 2821603
3274137 3490200 3701739 3906936 | 3
4 2825484 3054917
3277800
3494462
3705205 3910309
5 2829364 3058684 3281461 3498022 3708669 3913682
5
6 2833241 3062450 3285,121
3501580
3712131 3917052
6
7 2837117
3066214 3288778 3505137 3715592 3920421 7
8 2.840990 3069976 3292434 3508692 3719051 3923789
8
9
2844862
3073736 3296089 3512246 3722508
3927155
9
2848732 3077494 3299741 3515798 3725965 3930520 10
2852600 3081251 3303392
3519348
3729419 3933883 II
2856466 3085006 3397041 3522897 3732872
3937245 12
13 2860330 3088759
3310688 3526444 3736323 3940605 13
14 2864192 3092510 3314334 3529989 3739773 3943964 14
15 2868053 3096252
3327978 3533533
3743221
3947321 IS
16 2871911 3100007 3321620 3537075
3746668
3950677 16
17 1375768 3103752 3325261, 3540615 3750113
3954031 17
18 2879622
3107496 3328900 3544154 3753357 395738+18
I9
3111238 3332537 3547691 3756999
3960735 19
2887326 3114979
3336172 3551227 3760440 3964085 20
2891175 3118717
3339806
3554761 3763879 3967434 21
2895022 3122454
3343438 3558293 3767316 3970781 22
23 2893867 3126189
3347008 3561824
3770752 3974126 23
24 2902711 3129922 3350697 3565353
3774186
397747024
25 2906552 3133054 3354323 3568880
3777619
3980813 25
26
2910392
3137383
3357949
3572406 3781050
3984154 26
29.14229 314IIII
3361572 3575930 3784480
3987493 27
28 2918065 3144837 3365194 3579453 3787908
3990831 /28
29
_2921899 3148561 3368814 3582974 3791335 3994168 29
30 2925731 3152284 3372432 3586494 3794700 3997503 30
31 12929561 3156005 3376049 359001I 3798184
4000837 131
2933390 3159724 3379664 3593528 3801606 4004169 32
33 2937216
3163441 3383278 3597042 3805026 400750033
34
2941C41 3167156 3386889 3600555 3808445
4010829 34
2944863 3170870 3390499 3604067 3811863 4014157 35
2948684 3174582 3394107 36075 76 3815279 4017484 36
2952503 3178292 3397714 3611084 3818693 4020809 37
2956920 3182000
3401319 3014591 3822106 4024132 138
39 2960136 3185706 3404922 3618036 3825517
402745439
2963949 3189411 3408524 3621599 3828927 403077540
2967760 3193114 3412124 3625101 3832335 4034094 41
2971570 3196815 3415722 3628001
3835742 403741242
43 2975378
3200515 3419319 3632100 3839147 4040728 43
44 2979184 3204213 3422913 3635597 3842551 4044043 44
45 2982988
3207909 3426507 3639012 3845953 4047356 45
46
2986700 3211603 3430098 3642586 3849354 4050068 46
47 2990591 3215295
343.3688
3640079 3852753 4053978 47
1.2994389 3218986 3437276 3649569 3856151 4057287 48
49 2998186
3222675 3440863 3653058 3859547 4060595 49
$9:3001981 3226362 3444448 3050546 3862942 4063901 50
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3009565 -3233731
3451612
3663516 3869727
4070509 52
53 3013355 3237413 3+55192 3666999 3873117 4073811 53
54
1
3017142 - 3241094 3458770 3670480 3876506 407711I 54
55 3020928 3244772 3462347 3673959
3879893 4080410 55.
3074712 3248449 3465922 3677437
3883278 4083708 56
57 3028494 3252124 3469495 308091. 3886662
4087004 57
58; 3932274 3255797
3473067
3684389 3890045 4090298 58
593036052: 325946 3470637 3087862 3893426 4093591 59
60: 30 39829 3263138
3480205 3691334 3896800 4096883 62
27
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32
در کر دیا د ت تر
35
36
37
38
40
41
42
|
+
48
51,
52.4
A
56
84 A Table of Logarithmic Verſed Sines.
M, 42 Deg. 43 Deg. 44 Deg. 45 Deg. 46 Deg. 47 Deg.
M
ŽIO
I
46701421
4673190
2.
00V auto heed O
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4682327
4685370
с сочети бы ўно
5030806
II
4333306
16
17
18
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W
4349316
4724816
.
20
21
22
1
27
08.4091383 9.42918399.44818086.4667093 -9.4847860 9.50242994
4100174 4295015
4484934
4350836 5027198
4103462 4.290220 4488059
4853810
50301021
2
3 4106750
4301424 4491183 4676237 4856783 5033005
3
4 4110030 4304026
4494305 4679233 4859755
5035906 4
5 4113321 4307827 4497426
4862726
5
6 411660 4311027 4500546
4865696 GO4 1705
4119885 4314225 4503664 4688412 4860604 5044603
7
8 4123160 4317422 4506781
4691452
4871631 5047500 8
୨ 4126445 4320617 4509897
4694492
4874507
5050396 9
IO
4129722
4323811 4513011 4397530 4877562 5053240 IO
II
4132998 4327004 4516124 4700566 5830525 5050483
12 4136273
4330196 4519236
4703602
4883488 5059076 12
I3 4139546
4522346 4706636
4886449 5061967 13
14 4142818
4336574 4525456 | 4709669 4889409
5064857 14.
15
4146088
4339762 4528564
4712701 4892368
5067745 15
4149357 434294
8
4531670
4715732 4895326 5070633 16
4152625 4340135 4534776 4713761 4898282 5073519 17
4155891
45378801 4721789 4901237 5076.05 18
19 4159156 4352498 | 4540982
4004191
5079289 19
4162419
4355678 4554084 4727841 49071.14 5082172 20
4165081 4358858 4547184 4730866 49100y6 5085054 21
4168942 4362036 4550283 4733889 4913046
5087934 22
23 4172201
4:65212
:
4553380 4730211 4915996 5090314 23
a 4 4175459
4308387 4556477 4739932
4.)18944 5093693 24
25 4178715 4371561 4559572 4742951 4921891 j096570 25
26
4181970
43747741 4562665
4562665 1 4745969 4984836 5099446 26
4185223
4377905 4565758: 4748986 4927731 5102321 27
28
4188475
4381075 4568649 475 2002 4930724
SIO¢195 28
29 4191726 4384243 4571939
4755016 4933667
SIC8008 29
4194975 4387411 4575027 4758030 4936608
ST10940 30
31
4198223
4390576
45781154761042 49,39547 5113810 31
32
4201470 4.3937414581201 į 4764052 4942486
5116679 32
33
4204715
4396904 4584286 4767062 1945-24 5119548 33
34
4207959
440CCOO 4537369 4770070 4948360 5122415,34
35
4211201 4403227 4590451 4773078
4951295
512528: . 35
4214442
4400366
459,3532 4776083 4954229 5128146, 36
4217681
4405544 4590612 4779088 4957162 513100937
4220920 4412700 4599690 4782092 4960093 5133872 i 38
32 42241561 4415855 4602767
4785994
4963024 5136734 39
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4419099 4605843 4-83095 4965953 513952440
4230626 44-27021 4608918 4791095 4968881
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45 4243548 i 44,34753 4621203 4803082 4980; 8% 5157879 1 45
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4648771 4829981 5006040 5179521 54
55 4275756 4466158 4051828 4832964 5009752 $182365155
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4282181
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4657938 4838926 5015572
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5018480 5 I9o888 38
39
428800I 4478681| 4661043 4844883 5021388 5193728' 59
60 42913094471808' 4657093 4847860 5024294 $196566 : 00
30
36
37
38
6४४४
41
42
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56
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57
En
58
1
A Table of Logarithmic Verſed Sines. 85
0
I
II ان کی
I
2
2
1
Іо
II
I
I 3
14
IS
17
21
22
5914162
M 48 Deg. 49 Deg. 50 Deg. 51 Deg. 52 Deg. 53 Deg. M
9:5196566, 9.5364839 9.5529265 9.5689987 9 5847139 9.6000849
5194403
5531974 5692635 5849729 6003382
5202739 5370321 5534681
5695282 5852318 6005914
3 5205073 5373151
5537388
5697928 5854905 6008446 3
4
5207907 5375919 5540094 5700573 5857492 6010977 4
5 5210739 5378686 5542798 5703218 5800078 6013506 5
6
5213571 5361452 5545502 5705861 5862663 6016035
6
7 5210401 5304218 5548204 -5708503 5865247 6018563 7
8
5212930 5386982 5550906 5711144 5817830 6021090
8
9 5222058
53*2745
5553606 5713784 5890412 6023616 9
5224835 5392507 5556306 5716423 5872993
602614110
5227711 5395263
5559004 5719062 $875573 6028665 II
5230536 5398027 5561701 5721699 5878153 6031188 12
5233379 5400786 5564398 5724335
5830731 603371013
5236182 5403544 5567093 5726970 5883308 603623214
$239003
5406301 5569787 5729605 5845885 6038752 15
16
5241823 5409056 5572481
5732238 5388460 604127216
5244043 5411811
5575173 5734870 5891034 6043791 17
18 5247461 5414564 5577864
5737502 5893608 6046308 18
19 5250278
5417317 5580555 5740132 5896181 6048825 19
20
5253094 5420068 5583244
5742761 5848752 605134120
5255900 5422818
5585932 5745390 5901323 605385021
5258722
5425568
5538619 5748017 5903893 6056370 22
23 5261535
5428316
5591305 5750643 5906461 6058883 23
24 5264346 5431003 5593991 5753269 5909029 6061396 24
25 52671571 5433809 5596675 5755893 5911596
6063907125
26 5269900 5436554 5599358 5758517
606641726
27 5272774 5-439298 5602040 5761139 5016727 6068927 27
28
5275582
54-2041 5604721 5763701 5919291 6071436 28
29 5278388
5444783 5607401 5766382 5921854 607394329
30
5281193 5447524 5610080 5769001 5024417 607645030
5283997! 5450264 5612759
5771620 59269781 6078950 31
5286799 5453002 5615436 5774237 5929538
6081401 32
5289601 $455740 56181125776954
5932098
6083965 33
34 5292402 5458477 5620987 5779470 5934656
605040834
35
5461212 56234611 5782085 5937214 6088971 35
5298000 5463947 5626134
5794698
5939770 60y1472 36
37 5300797 5406681 5628806 5787311
5942320
6093972 37
5303594 5469413
5631477 5789923
5944881
6090472 38
39 5306,389 54,72145 5034147 5792534 5947437 609897139
40 5309183 5474875 56.36861
5795144 5949907 6101469 149
41 53119701 5477604
5639484 5797753 59525.39
6103965 / 41
42 53147681 5480333 5642151 5 800361 SU55090
6IC646142
43 5317559 5493000 5644817 5802968
5957640
6108956:43
44 5320344 5485786 5647482
5805574
5960189 611145 I44
45 5323137 5488511 5650146
5808179 5962737 6113944'45
46 5325929 549126 5652809 5810783 5965285 (116436 46
47 5328712 5423959 5655471 5813380 5967831 6118928 47
48 5331497 5490081 5658132 5815988 5970376 6121418 48
42 53.34281
5499402 566079% $818589 5972921 6123908.49
5337065 5502122 5663451 5821190 5975464 6126397 : 50
51
5339847 5504841 5666109 5823789
5978007
6128885151
52
5507559 5668766: 5826387 5980549 613137252
53 5345468 5510276 5671423 5826985 5983089
54 5348137
5512992 5674078 5831581
5985029
6136343 1 54
55 5350965 5515706 5676732 5834176 5989168
613882755
5353742 5518420 5679385 5836771
5990706
6141311 56
57
5521133 5682037 $839364 5993243
6143793 57
58 5359293 5523845 5684688
5841957 5995779 6146275 58
59 53020671 5526555 5687338 5844549 5998314 6148755 159
63 5304839 5529265 5689987. 5847139
6000849
6151235 60
31
32
33
5295201
در در در در دره دی در
36
38
1
50
5342628
6133858 53
56
535010
86 A Table of Logarithmic Verſed Sines.
M; 54 Deg. 55 Deg. 56 Deg. 57 Deg. 58 Deg. 59 Deg. M
I
2
cov, an AW NO
II
6480394
I 9
6919302 / 28
0 9.6151235 9.62984121.6442486..6583558 9.6721725 9.6857076
6153714 6300838 6444861 6585884 6724003 6859309
6156192 6303264 6447236 6588210 6726281 6861541
3 6158669 6305688 6449610 6590535! 6728558
6863772
4
6161146 6308112 6451933 6592858 6730835 6866002
5 6163621 6310535 6454355 6595182 6733110 6868231
6 6166096
6312957 6456726 6597504
673.5385 6870460
7
6168569 6315378 6459097 6599825 6737659 6872688
8 6171042 6317799 6461407 6602146 6739932 6874915
6173514
6320218 6463836
9
6604466 6742205 6877142 9
6175985
10
6322637 6466204 6606785 6744476 687936810
6178455 6325055 6468571 6009103 6746747 6881593 11
6180924
12
6327472 6470937
6611421 6749017 6883817 | 12
13 6183392 6329888 6473303
6613737
6751287 68860413
6185860 6332303
14
6475667 6616053 6753555 6888263 | 14
6188326
15
6334717 6478031 6618368 6755823 6890485 | 15
16
6190792 633?131
6620083 6758090 6892706/16,
6193256 6339543
6482757 6622996 6760356 6394926 17
18
6195720 6341955 6485118
6625309
6762622 689714618
0198183 6344366
6487479
6627621
19
6764886 689936519
6200645 63467766489839
6629932
20
6767150 0901583 / 20
6203107 6349105
21
6492197
6632242 6769413 6903801 21
6205567 6351594 6994556
6634552
22
6771676 6906017 22
6208026! 6354001 6496913 6636860
6773937
23
6908233 23
6210485 6 6356408; 6499269
24
6639168 6770198 6910449 24
6212943 6358814
25
6501625
6641475
6778458 6912663 25
26
6215400 6361219 6503980 6643781 6780717
6914877 26
6217855 6363623 6506334 6246087
27
6782976 6917090 27
28 6220311 6366026 6508687
6648392 6785234
6222765 6368429
29
6511039 6650696 6787491 692151-3 29
6225218 6370830
30
6513391
6652999
6789742 16923724 | 30
6227670 6373231 6515741
6655301 6792002 6925934-3,1
6230122 6375631 : 6318092 6657603 6794257 6928143 32
6232573 6378030 ; 6520441
6659903
6796511 6930352 33
6235022 6380428 6522789 6662203 6798764 6932559 34
6237471 6382825 6525136 6664502 6801016 6934766 35
6239919 6385222: 0527483 6666801 6803268 6936973 36
6242367 6387618: 6529829 6669098 6805519 6939178 37
6244813 6390012 6532174
6671395
6807769 6941383 38
6247258 6392406 6534518
6673691 6810018 9943587 39
6249703 6594800 6536861 6675986 6812266 694579040
6252147 6397192 6539204 6678281 6814514 6947993 41
6254589 6399583 6541546
6680574
6816761 6950194 42
6257031 6401974 6543887
43
6682867
6952396' 43
6259473 6404364 6546227
44
66851596821253 6954596 ; 44
6261913
6406753
6548566
6687450
6823498 6956795 : 45
6264352 6409141 6550904 6689741 6825741
6958994 46
6266791
0711528
47
6553243 6692030 6827985 6961192' 47
6269228 6413914 6555579
6694319 6830227 6963399 48
6271665 6416300 6557915 0696607 6832469 6965,586 49
6274101 6418685 6500250 6698895
6834710 6967782.50
6276536 6421068 6562585 6701181
6836950 6969977-51
6278970 6423452
52
6564918 6403467 6839189 6972177:52
6281403 6425834 6567251 6705752
6841428 6974365 53
6283836 0428215 6869583 6708036 6843665 6976558-54
6286.67 6430596 6571914
6710319 6845902 6978750-55
6288098
6432975
6574245 6712602
6848139
6980942'56
6291128
6435354 0576574 6714884
6850374 6983132 57
6293557 6437732
9578903
6717165 6852609
6985322458
6295985 6440109 6581231
6719445
59
6854843
698751-2; 59
60 6298412). 6442486 65835581 6721725 68570761. 6989700 - 60
ON
31
32
33
34
35
36
37
38
ب با دا
39
40
41
42
681-9007
45
46
48
49
50
51
11
53
54
55
56
57
الدم
58
5
A Table of Logarithmic Verſed Sines. 87
I
2
IO
II
12
>
13
21
22
7421286
9662040 24
M 60 Deg. 61 Deg., 62 Deg. 63 Deg. 64 Deg. 65 Deg. M
.
i
9.6989700 9.7119677 9 7247087, 9.7372002 9.7494494 / 7.76146300
6991888 7121822 7249189 7374063 7496516 7616613 I
6994075 7123965 7251290 7370124 7498536
7618595%
3 6996271 7120108 7253391 7378184 7500556
7620577 | 3
4 6998447 7128250 7255491 7,380243 7502576
7622557 4
S.
7000631 7130392 7257591 7382301 7504595
76245375
6 7002816 7132533 7259689 7384359 7506613 -76265176
7 7004999 7134673 7261787 7386416 7508630 7628496 7
8 7007182 7136812 7263885 7388473 7510647 76304748
9 7009363 7138951 7265981 7390529 7512663
76324529
7011545 7141089 7268077 7392584 7514679 7634429 10
7013725 7143226 7270172 7394638 7516694 7636405 II
7015905
7145362 7272267 7396692 7518708 763838112
7018084 7147498 7274361 7398745 7520721 7640356 13
I4
7020262 7149633 7276434 7400798 7522734 764233014
15 7022439
151768 7278546 7402850 7524746
764430415
16 7024616
7153901 7280638 7404901 7526758
7646277 16
17 7026792 7156034 7282729 7406951 7528769 7648250 17
18 7028967 7158166 7284820 7409001 7530779 765022218
19 7031142 7160298 7286910 7411050 7532789
7652193 19
20 7033316 71624.29 2288999 7413099 7534798
765416420
7035489 7164559 7291087 7415146 7536806
765613421
7037661 7166688 7293175 7417193
7538814
7658103 22
23-
7039833 7168817 7295262 7419240
7540821
7060072 23
24
7042004 7170945 7297348
7542828
25 7044174 7173072 7299434 7423331 7544833 766400725
26 7046344 7175199 7301519 7425375 7546839 7665974 26
27 7048513 7177325 7303603 7427419 7548843
766794027
28 7050681
7179450 7305686 7429462 7550847 766990628
29
7052848
7181975 7307769 7431505 7552850
7671871 29
30 7055015" 7183698, 7309852 7433547 7554853
31 7057180 7185821
7311933
7435588 7550855
767.5799
7059346 7187944 7314014
7437628 7553856
7677762 32
33 7061510
7316094 7439068 7560856
767972533
34 7063674 7192186 7318174 7441707
7560850
35 7065837 7194307 7320252
7443746 7564856
908364835
36
7067999 7196426 7322331 7445704
7566854
1685608 36
7070160 7198545 7324408 7447821 7568852 1687568
38
7072321 7200603 7326485 7449857 7570850
368952838
39 7074481
7202781 7328561 7451893 7572847
1691486 39
40 7076641 7204898 7330636 7453928 7574843
41 7078799 7207014 7332711 7455963 7576838
7695402 40
42
7080957 7209129 73.34785
7457997 7578833
7697359 42
43 7083115 7211244 7336858 7460030 758c827
7699315| 43
44
7085271 7213358 7338931 7462063 7582821
770127144
45 7087427 7215471 7341CO3 7464095 7584814
7703225 45
7089582 7217584 7343074 7466126 7586806 770518046
47 7091736 7219695 7345145 7468156 7588798
7707134 47
48
7093890 7221807 7347215 7470186 7590789 7709087 48
49 7096043 7223917 7.349284 7472216
7592779 7711039 49
50 7098195 7226027 7351355 7474244 7594769 7702991/50
51 7100346 7228136
7353421 7476272 7596758 771494250
52 7102497 7230244 7355488 7478299 7598746
771689352
53
7o4647 7232352 73575.55 7780326 7600734
7718843 53
54
7106797 7234459 7359621 7482352 7602721 7720792 54
55 7108945 7236565 7361086 7484377 7604707 7722741 55
56
7111093 7238671 7363750 7486402 7606693 7724689 56
57
7113240
7240776 7365814 7488426 7608679 7726,636 57
58 7115387 7242880 7207878 7490449 2110663 7728583 58
7117532
7244984 7369940 7492472 7632647 7730530.59
60, 7119677
71196771 7247087 7372002 7494494 2014630 ī 1324751 60
7673835 30
31
32
7100C66
768168734
37
En Go w Was
37
7693444 40
46
59
88 A Table of Logarithmic Verſed Sines.
M 66 Deg. ,67 Deg. 68 Deg. 69 Deg., 70 Deg. 71 Deg.
M
I
00 am a Nord O
AN CNC - M = 0
IO
II
12
16
7703521
8102197
7765457
7993288
Q
19
20
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21
22
23
I
8223475
25
8003173
81204459
A A N N N A
09.7732475).7848090 9,7461533 1 9 8072860981821269.8289341
7734420 7819998 7963406 8074698 8183230 8291152
7736305 7851906 7965273 8076536 8183733 8:92922
3
7738302 7853813 7467149
8078372 8187536 8294692 3
4 7740252 7855720 7069020 8080208 8189533 8246461 4
S 7742 194 7857626 7970890
8082044 8191140 {zy8229 5
6 7744136
7859531
7972760
8083879 8192941 8299997
6
7 7746077 7361436 7974629 8085713 8194742 8301765 7
8 7748-18
7863340 7976-198
8087547
$196542 8303532
8
9 7749958 7865243 7978366 8089380 8198311 8305229 9
7751098 7807140 7,80233
8091213
8200110 830700410
7753536 78690.19
7082100
8093045 8201938 8308830, 11
7755775 7870950
7933966
8094877 820.37.36 8310595 ( 12
13 7757712 7872852 7485832
8096708 82055.33 8312359 13 i
14
7759649
7874752 7987697 8098538
82073,30
831412314
15
7761586 7876652 7989561 8100362 8209120 831588615
6
7878551 791425
8210922
5317649 16
17
7880450
8104020 8212717 831141117
18 7707341 7882348
7995151 C105854 821.511
8321172 18
7769325 7881246
7797013 8107632 8216305 8322).33 19
:
7771258
7880143 7998873
8213519 8324024 20
7773191
7088037 80007.35
CL11335 8219391 83264541 21
7775123
7889935 8002546 8113101 8221084 8328213122
7777055 7891830 8004456 8117986
8329972 23
24 7778985 7893725
8006315
8116211 8225266
8331731 24
7780916 7895618
8118035 8227057
833348825
26 7782345 7897512
80500;I
8:20847 833524026
27 7784774 7099405 801183)
8122202 82-30636 8337002 / 27
28
7786703 7901297 8013746 8124104
8232425 8338759 28
29 7788630 7903188
8013602 8125926 8234214 8340514 29
30
7905079 8017453 8127748 8236002 8342269 30
31 7792484 7906970 8017313 8129569 8137789 8344024 31
7794410 7708859 8021167 8131389 8230576 8.345778 32
33 7796335 7910749
8023021 8133200 821362 8317532 33
7796260 7012673 8024875
34
8243147
83499285 34
35 7800184 7914525 8026728 81;63.0 8241,321 8351037 / 35
7802108 7910413 8028580 8138664
83527836
37
7804031 7918300
80,30432
8140481 824.750I 835454037
7305953 7920186 8032283 81.32208
8250284 835629138
39
7807875 7922071
8034133
81.4411
822,3067 835304139
40
7809796 7923956
8035933
825,36491 835979130
7811716 7925841 8037332
82.55631 8361540 41
7813636 7927725 &039631 8149560 8257412; 8363289 42
43
7815555 7424608
8041529
8259193 i 8305037 43
7817474 793171
80443377
8153187
41
8260y7,3
8305785 44
19392 7233373
8045224
815.5000
45
126127.53
9368532 45
7821,09 7935254 8047070 8350812 8261532
83;027346
7823220 7937135 8048976 8158024 8266312 8372024! 41
7825143 7939015 8050762
8160435
8:60038 837:770 48
7827058 7940895
49
8052606 2160246 8269866 8375515 1 49
7828973
50
7942714 8054451
821, 1622 0377259 30
7830883 7944653 8056294 8165866 8273412 8372003 51
7232301 7446531
8058137
8275-594
838.9746! 52
7034713 7948408 8059980
8169-173 8-76970 4382489 , 53
7836627 7950285
8061821
82-87-4-1 8.384231 : 54
7838832
2952161
8063663
8172053 82vi13
830597355
56 7840450 7254037
8274905 8252292 836771450
7872701 7955912
8176711 820405
8389455 157
7844271
7957786
8069183 8178516 8265837 8395195 153
7954060
59
8071022
8130322 Uz6:10) 837293515)
60
7961533 8072860 81321201 823) I
8304674 to
7790558
32
1
8135027
30
8246717
38
8145930
81.7745
41
42
815137-+
46
47
48
SI6wf?
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SI
52
menemu
816; -75
53
SL
5171291
551
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8067344
57
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فروت اور
در
A Table of Logarithmic Verſed Sines.
89
M 72 Deg. 33 Deg. 74 Deg. 75 Deg. 76 Deg. 77 Deg. ,M
1
2
lomam ex bono age
сло
IO
IO
II
II
8417233
14
Qas
FI IIIIII
19
25
I
20
21
22
22
4741881
29
30
دي کي کي د کم و در دره ن د ن در دره ده
09.8394674 , 9.8498052 9.85995609.8699243 9.8797140 | 9.8893291
8396412 | 8499759 8601237 8700889 8798756 8894879
8398150 8501465 8602912 8702534
8800372 8896467
8399838 8503771 8604588 8704179
3
8801988 8898054
8401625 8504877 606262 8705824
4
8803604 8899640
8403361
5
8506582, 8607936 8707468 8805218 8901226
6
8905097
8508280 8609610 8709112
8906833 8902812
8405832
7
8509990 8611283 8710755
8608446 8904397
8 8408567
8511693 8612956
8810060
8712398
8905982
8410301
8513306 8614628 8714040
8811673
9
8907566
8515022
8412035
8616300
8715632
8813285 8909150
8516800
8413768
8617971
8717323
8814897 8910733
8518502
I 2
8415501
8610642 8718963 8816508 891231612
8520203
13
8621312
8720004
8818119 8413898 13
8418965
8521903 8622981 8722243
8819730 8,1548014
15
8523603
8420696
8624651
8821340
8723883
8917061 | 15
16
8525302
8422427
8626319
8725521
8822949
891864216
8527001
17
8627987
8824558
8424157
8727159
8920222 17
18
8528692
8425886
8629655 8728797
8826167 8921802 18
85303.)
8427615
8631322
8730434
8827775
892338219
8532094
8429344
8632989
8732071
8829382 8924961
8533720
8431072
8634655
873.3707
8830989 8326539
85.3$486
8432799
8636320 8735343
8832596
8928117
23
8537182
8434526
3637985
8834202
8736978
8929695 23
8538877
2.4 8436252
8639650 8738613 8835807 893127224
8540572
8740248
8641314
8837413
25
8932849
8437978
25
26
8542266
8439703
86412978
8839017
8934425 26
27
8441428 8543959 8644641
88.40621 893600027
8743515
28
8545653 8646303 8745147
8443152
8842225
8937576 28
8547345
8444876
8647966 8746780 8843828 8939150 29
8549037
8446599
8649627 8748412 88.45431 8940725 30
8150729
448322
8651288 8750043
8847033
8942299 31
32
8552420
8450044
8652949
8848635
8751674
894387232
33
8554110
8451766
8654602
8850236
8753304
894544533
34
8555800
84534.87
8656262
8851837
8754934
8947017 | 34
8557490
35 84.55207
8657923 8756563 8853438 8948589 | 35
36
8559179
8456927
8052586
8355038
8750192
8950161 36
E560567
8754821 8856637
8458647
8951732 37
8562555
8.60366
8662902
8761.449 8858236 8953302 38
85632-42
8,63076 8859834
39 8462004
8954872 39
8565922
8463802
8666216 8764703
89614132
8956442 +2
8567616
8465520
8667872
8760329
8863030 895801141
8569302
8407237
8669,527
8767955
386.1527 895958042
43
8570987
8468953
867182
8866223
8769581
8961148 43
8572072
44
8470609
8672837 8771206
8867219 896271647
8574356
45
8472334
867-621 8772830 8969415 896428315
46
8576040 8676115
8,72454
8474099
88.ICIO 896585046
47
8577723
875013
8677-98 07
४४:2605
;;60:3
8967416 47
8579406
8477527
8679450
072770I
237199
896898248
49 6479240
8581089 8681102
8970547 49
8779374
8582770
8480953
8082754 878046 cili; 306 897411250
8544452
868 dec 5
SI
8762567 1879978
482605
8973677 51
8586132
52
E6 8boju
4995 I
8474377
8,54188
I 897524152
53
E567813
8186088
8687706
COD:162
8785809
897680453
$4
8509492
8407799
86819355
883,3754
$787429
897836754
55
8591172
09509
8691004
8985344
87 Syosh
897993055
56
8592851
8.391219
8790668
888935
8981492 56
8594529
8492928
8694301 8792266 8:88525 8983054 | 57
SOI14
58 8494636 85960068697949
3793905
898401558
8496344 8597884 8697596 $795582 ES91703
59
83861761 59
8498052 8599560! 8699243 8797140*8893291 8987736 66
K
31
8661244
37
38
8664559
42
40
41
42
I
切
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to ho ho
6,975742
50
809653
57
60
90 A Table of Logarithmic Verſed Sines.
I
2
cov.an to ww.no
!
II
II
I3
고 ​· 11; if 1
I5
med to No
8284799
18
20
21
2I
22
M 78 Deg. 79 Deg. 80 Deg: 81 Deg. 82 Deg. 83 Deg. M
698987736 12.9080510 9.9171650
098987736 9.9080510 9.9171650 9.9:61188 6.93491581 99435591
8989296 9082043 9173155 9262667 9350611
8990855 9083575
9437012
9174660 0264646 9352064 9438446
3 8992314 2085106 9176165 9265624
9353516 9439873
4 8993973 9086637 9177669 926710r: 9354968 9441300
5 8995531 9088167 9179172 9268379
9356419 9442726
8997088 2089697 9180675 8270055
9357870 9444151
8998645 9091237 9182178 9271532 9354321 9445577
8 8990202 9092756 6183680
9273008 9360774
9447001
9 9001758 9094285 9185182 9274483 9362220
9448426
IO 9003313 9095813 9186683
9275958 9303670
9449850
10
9004869 9097341 9188184 9277433
9365119
9451273
I 2 2006423 9098868 9189685 9278907 9306567 9452696 | 12
9007978 9100395 2191185 9280380 9368015 9454119 13
14 9009531 9101921 9192684 9281854 2369462
2455541
14
9011085 2103447 9194183
9283327
9370909 9436963 | 15
16 2012638 9104973 9195682
9372356 9458385 16
17 9014190 9106498 9197180 9286271 9373002 945980617
9015742 9108022 9198678
9287743
9375248 9461226 | 18
19 9017294 9109547 9200175 9289214 9376694 9462646 19
9018845 QI11010 9201672 9290684 9378139 9364066 20
9020395 9112593 9203169 9292155 9279583 9405486
9021945
Q114116
9204665 9293624 9381027 9406904 | 22
23 9023495 9115632 9206160 9295094 9382471 946832323
24 9025044
9117161 9207656 9296563 9383914 9469741 | 24
25
9026593 9118682 9209150 9298031 9305357 9471159 | 25
26
9028141 9120203 9210644 9299492 9386800 947257626
27 9029689 9121723 9212138 9300967 9388242
9473993 27
28
0031236 91232441 9213632 9302434 9389683
9475409 | 28
29 9032783 9124763
9215125 9303901 9391124 947682529
30 9034330 9126282 9216617
9305367
9392565
947824130
31 1 9035876
9127801
9218109 9306833 9394005 9479656 31
37
9037421 9129319 9219601 9308299 9395445 9481071 32
33 9038966 9130837
9221092
9309764
9396885 9482486 1 33
34
9010511 9132355 9222583
9398324 9483899 | 34
35 9042055
9133872
9224073 9312693 9399762 9485313 35
36
9043599
9135388
9225563 9314156 9401201 9486726 36
37 9045142 9136904 9227052 9315620 9402638
9488139 37
38
9046685 9138470 9228541 9317083 9404076 948955138
39 9048227) 9739935 9230030 9318545 9405513 9490963 39
40 9049369 9141450 9231518 9320007 9400949 9492375 40
41
9051310 5142964 9233006 9321469 9408385 949378641
4.2
9052851
9144478
9234493 9322930 9409821 949519642
43 9054392 9145991 9235980 9324391 9411256
44 9055932 9147504 9237466 9325851
9412691 949801644
45 9057471 9149016 9238952
9327311 9414126 94994.26 45
46
9059011 9150528 9240437 9328771 9415560 950083546
47
9060349
999 2040 924 1922 9330230 9416993 9502243 47
9062087 2153551 9243407
9418426 950365248
49
9063625
9155062
9244891
9419459 950505949
50 9065163 9156572 9246375
9421293
9506467 50
9066692
9156082
9247858
9422723
9507874 | 5!
52
9068236 9159591 9249341
9424155 950928052
53
9069772
9161100 9240824 9338975
9425586
951068653
54 9071307 9162609 9252306 9340431 9427016 951209254
55 9072842
9164119 925378
9341887
9428447 951349755
9074377
9165624 9255268
9343342 9429876 9514902 56
57 9075911 9167131 9256749 9344797 9431306
9516307 57
9077445 9168638 9258229 9340251
9432735 951771158
59 9078978 9170144 9259709
9347705 9434163
951911552
'రం, 1080, 10
9261188
9435591 9520518 60
دن کا کا م درا به در
no a
9311228
No a
9496607 43
48
9331688
9333146
9334604
9336062
9337518
SI
56
Слля
58
9171650
93491581
1
{
.
A Table of Logarithmic Verſed Sines. 92
London
LOW
O-H A na No a
678
1
1
9933808
IO
II
1
14
1
20
21
22
23
M 84 Deg. 85 Deg. 86 Deg. 87 Deg. 88 Deg. 89 Deg. M
9.952051819.9603967 9.9685967 | 9-9766544 9.9845725 9.9923536 !
!
952192] 9605345 2687321
9767875 9847033 9924821
9523323
2606723 9688675
9769206 9848341 9926106
9524725 9608101 9690029
9770536 9849648 9927391
3
4 9526127 9609478 9691382
9771866
9850955 9928675 4
5
9527528
2610855 9692735 9773195 9852262 9929959!
9528929 9612232 9694088 9774525 9853568 9931243
6
9530329 9613608
9695440 9775853 9954873
9932526
7
9531729 9614983 9696792
9777182
9856179
9533129 9616359 9698143 9778510 9857484
9935091 | 9
9534528
2617733 9699494 9779837 9858788 9936373 | 10
IO
9535927 9619198 9700845 9781164
9860093 9937654 | 1!
*9537325 9620482
9702195 9782491
9861396
9938936 12
13
9538723 9621856 97035 45 9783818 9862700 9940217' 13
9540120 9623229 9704894 9785144 9864003 9941497 1 14
15
9541518 9624602 9706243 9786469 9805306
994277715
16 9542914
9625974 2707592 9787795 9866608
9944057 16
17
9544311 9627346 9703940 9789119 9867910 9945336 | 17
18 9545706 9628718 9710288
9790444 9869211
9940615 18
19 9447 [02 9630089 9711635
9791768
9870513 9947894 12
9548497 9631460 9712982 9793092 9871813
9949172) 20
9549892 9632830 9714329 9794415 9873114
9950450 21
9551286
9634200 9715675
9795738
9874414
9951728 22
9552680
9635570 9717021 979706!
9875713
9953005 | 23
24
9554073 9636939 9718367 9798383
9877013
9954282 24
25
9555466
9638308
9719712 9799704 9878312 9955558 25
26 9556859 9639676
9721057
9801026
9879610
9956834 26
27
9558251
9641044
9722401 9802347 9880908 9958110 27
28 9559643
9642412 9723745 9803668 9882206
995938528
29 9561034
9643779
9725088 9804988
9883503 9960660.29
30 9562425
9645146
9726431 9806308 9884801 9961935 ! 30
9646513 9727774 9807627 9886097 9963209 | 30
3%
9647879 9729117 9808946 9887393 9964483 32
33
9649244
9730458 9810265 9888689 996575633
34
9567985 9650610
9811583 9889985 9967029 34
35 9569374
9651974
9733141 9812901 9891280 9968302 35
0's 70963
9734482
9814219 9892575 9969574 36
37 9572151
9654703
9735822
9815536 9893869 9970846 37
9573539
9656067
9737162
9810853 9895163 9972118 38
39
9657430
9738502 9818169 9896457 9973389 39
40
9576313
9658793
9739841 9819485 9897750 9974660 30
41
2660155
9741180 9820801
9899043 997593141
42
9661517
9742519 9822116 9900335 9977201 42
43
9580471 9662879 9743857 9823431 9901627 9978471 43
44 9581857
9664240 9745194 9824745 9902919 997974044
45 9583242 9065601 9746532 9826060 9904210 9981009 | 45
46
9584626
9666961
9747868
9827373
9905501 9982278 | 46
9586010
2668322 9749205 9828687 9900792 9983546 +7
48 9587394 9069681 9750541 3830000 9908082 9984814 48
49 9588777 9671041 9751877 9831312 9909372 998608! 49
50 9590160
2672399
9753212 9832624 9910662 9987348 30
9591543 9073758 9754547 9833936 9911951 998861551
52
9592925 9675116
9755882
9835248 9913240 998988252
53 9594306
2076474
9757216 9836559
9914528
9991148 53
54
9595688
9677831
9758550 9837869 9915816
9992414 54
55
9597069
9679188 9759883 9839180 9917104 9993679 55
9598449
9680544 9761216
0840490
0918391
9994944 59
57
9599829 9681901 9702549
9919678 9996208 57
9601209 9683256 9763881 0843108 9920964
999747358
59 9602588 9684612 9765213 9844417 9922250 9998737 59
69 9603967' 2685967 9766544 984572$ 9923536/ 10.000000169
!
31
9563816
9565206
9566596
9731800
36
9653339
دی. د بي نه د
38
9574926
9577699
9579086
47
51
09
56
2841792
58
>
82" À Table to convert Sexageſimals into Decimals to
every Second, and every 6 Third Minutes of the
Quadrant, do contra.
O
6
I 2
18
24
30
36
42
48
54
0
lo hay
0216 0233
2
2
I
OCQC
O as
1883
205
21
12
25
1 1 보보
​29
18
3416
35
21
21
22
4216
26
4516
4716
2
.0000 ,0016 0033.005
.0066.0083 01 0116,0133.015
I 0166 0183
02
025 0266 0283 03 0316
20333
035 0366 0383 04 0416
0433 045
04661 0983
3 95
0516 | 0533
055 0566 | 0583 06 0616 0633 065 3
4 0666 0683 07
0716
0733
075
0766 | 0783 08 0816 4
50833 085 0866 0833 09
0916 0933
095 0966 0983 5
YO16
1033 105
1366 1083 II
I116 | 1133
[
115
6
7 1066
7
1053
II
II16
II33 115
1166 1183 12 1216
7
8
1333
135
1360 | 1383 I4
1416 1433
145
1466 1483
8
9 I-5 1516
1533 155 1566 1583 16.
|
1616 1633 165 9
10
1666 1683 17 1716 | 1733 175 1766 1783 | 18
1816 10
II
1833 185
1866
19
1916 | 1933
195 1966 1983 11
122
2016
2033
2066 2083
2116 2133 215
13 2166 2183 22, 2216
2233 225
2266 2283 | 23 2316 13
14
2333 235
2366 | 2383 24
2416
2433 245 2466 2483 14
IS
2516
2533 255 2566 2583 26 2616 2633 265
IS
16 2066 2683 27 2716 2733
275 2766 2783 28 281616
17 2933 285 2866
2893
2916 2933 295 3966 2983 17
18
3 3016 3033 305 3066 3083 31
3116
3133 315
19 3166 3183
32
3216 3233 325 3266 3283 | 33
331619
20 3333 335
3366 3383
3383 1.34
3433 345 34661 3483 / 20
3516
3533 355 35663583 36 3616 3633 | 365
3666 3683 37
3716 3733
375 3766 3783 38 3816 1 22
23 3833 385
38663883
39
3916 | 3933
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