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I11

THE TRIGONOMETRY OF SMALL CURVILINEAR
TRIANGLES ON A SURFACE

THErst notions concerning plane and spherical trigo-
nomefitry date back to the Babylonians. The Greeks
related these notions to Menelaos' theorem on transversals;
substantial contributions have since been made by the In-
dians, by the Persians, and by the Arabs. One of these
last, Nasir Ad-Din, prepared in the thirteenth century a
treatise on pure trigonometry; that is, it did not deal di-
rectly with astronomy, though naturally it was influenced
by the needs of that science.

The Renaissance saw the completion of the deduction of
the relations, of which only three are independent, between
the six elements, the angles and sides, of a triangle; a t that
time, these relations were given in the extremely diverse

forms applicable t o the various cases arising in practice. It

is necessary t o come on t o Euler t o find the formule of
trigonometry in their actually compact form : previously,
for want of having defined the trigonometric functions in
such a systematic way that the definitions should hold for
all real values of the angle, one was obliged t o distinguish
a multitude of cases and subcases.

An important generalization of trigonometry, from the
theoretic point of view, was conceived by Gauss, who
broached the study of geodesic triangles; that is, triangles
formed by arcs of geodesics on an arbitrary surface. I n
this field appear researches of Christoffel, Weingarten, Dar-

boux', and M. Severi.2

206

Small Curvilinear Triangles 207

I propose now, as a further extension, the consideration

not merely of geodesic triangles, but of triangles formed by
arcs of arbitrary curves on an arbitrary surface. However,

to obtain relatively simple results, we shall restrict our-
selves t o a consideration of small triangles.

But first of all, just what is a m a l l triangle? If, in Eu-

clidean geometry, we are considering rectilinear triangles,
then smallness has no intrinsic sense, but depends on the
unit of length; we can always, by a suitable choice, repre-
sent the lengths of the sides by numbers as small as we wish.
T h e everyday criterion of smallness is determined here nec-
essarily by additional considerations, depending on most

varied circumstances. But, if the sides are curvilinear, there
exists a natural yardstick: it is the curvature. To fix the

ideas, let I' denote the maximum of the curvature for the

different points of the three sides, and let L denote the
maximum length of the sides. Then the product I'L is a
pure number, and we are perfectly well justified in saying
that a triangle is small if for it the above product is less

than a certain proper fraction, which we can treat as a
quantity of the first order.

Next, we are confronted with the problem of establish-
ing, as in ordinary trigonometry, the relations between the
lengths of the sides and the magnitudes of the angles of

the triangle. I n general, if the curves are arbitrary, we can

attain only functional relations, t h a t is, relations involving
all features of the functions defining the curvilinear sides;

but it is entirely different when we bear in mind that in

taking recourse t o the Taylor developments, we may utilize
the smallness of the triangle t o justify our considering as
negligible, relative t o unity, every quantity of order not
less than 1, or 2, or 3, etc.

Continuing with our generalities, we hasten to remark

208 Levi-Civita Lectures

that we can very well consider the curvilinear triangle as
being immersed directly in a containing Euclidean space,
or more generally in a containing Riemannian space of an
arbitrary number of dimensions; but it is convenient t o con-
sider first the case, which is undoubtedly the most interest-
ing from the geodetic or topological point of view, where
we suppose the triangle t o be drawn on a given surface u:
a plane, sphere, ellipsoid, etc.

If we take account of the above surface u, there is, ex-
cept for the plane, another element, the total curvature K

of the surface, which we must consider in order to appre-
ciate the smallness of a triangle. Here again there is a pure
number dependent on the dimensions of the triangle rela-
tive t o those of the surface: it is the product K L ~w,here K

denotes the maximum of K on the triangle and its interior.
We make the hypothesis that if I'L is a quantity of the

first order, then K Lis~of the second order; that is, is com-
parable with (I'L)2.

I n this lecture, we shall discuss first ($1) small plane
triangles whose sides are circular arcs, and shall outline in
the most straightforward way the exact trigonometric rela-
tions, writing however explicitly only first approximation
formula:; that is, we shall neglect terms of the second and
higher orders. This elementary introduction gives a first
orientation t o the analogous research in the general case of
an arbitrary surface and of a triangle with arbitrary curvi-
linear sides, when we shall take account of terms of the
second order but shall neglect those of order higher than

the second. It is easy t o see that, in order t o pass t o ar-

bitrary curves and to our ulterior approximations, we must
rely on a canonical representation of a curve in the neigh-
borhood of a fixed point on the curve. We take the op-
portunity of reviewing the deduction of the representation

Small Curvilinear Triangles 209

($2), and of obtaining the extension t o the case of curves
on a given surface. This development demands quite a bit
of preparatory material: intrinsic differentiation and for-
mulae of Frenet; expedient coordinates for a small region;
and parametric expressions ($3) for the coordinates of a
given curve on the surface. We study next triangles whose
sides are geodesics ($4);in this study we find ourselves led,
in the second approximation, t o spherical trigonometry,
either ordinary or Lobatchewskian ( $ 5 ) . We have now only
t o use the developments of the preceding sections t o ob-
tain, t o the same order of approximation, the trigonometry
of curvilinear triangles ($6).

I established the formulae for the first approximation in

1934,3as a rather remote corollary of an extended geometric
research concerning the mutual relations of three families
of curves traced on a surface. More recently, the late Cohn-
Vossen has shown4 t h a t we can obtain the same result by
direct elementary considerations.

I had done this on my own account, without, however,

publishing the remark, using the method which is presented
here in $1, and which moreover is obtained from the same
idea as that which inspired Cohn-Vossen. The passage t o

the second order, and eventually t o higher orders (QS), fol-

lows, on the contrary, as we have indicated, from an analytic
process admitting a systematic treatment.

I adjoin, for completeness, that $7 of this memoir con-

tains a slight remark relative t o the theorem of Legendre
on the resolution of spherical triangles.

I . SMALL PLANE TRIANGLES

FORMED BY THREE CIRCULAR ARCS

Let Ph, bePh+l, Ph+2 a plane triangle formed by three
circular arcs (Fig. l ) , with the obvious convention of re-

210 Levi-Civita Lectures

garding as identical those indices which differ by 3 or by a
multiple of 3.

1

Fig. 1

We fix as positive sense of describing the boundary of
the triangle, the sense which is defined by the increasing
order of indices, designated by the arrow in the figure. And
we represent by h the length of the side opposite the vertex
Ph, that is t o say, the length of the circular arc Ph+IPh+z;

by a h the length of the corresponding chord Ph+lPh+Z. If
Rh is the radius of the circle of which Ph+lPh+Bis an arc,

then the ratio 1 h / R h measures in radians the corresponding

central angle. It goes without saying, that for small tri-

angles, this angle is correspondingly small : in particular, we
may always take this angle as being acute.

I introduce further the curvature Y h taken with a proper
sign, having i/Rh for absolute value, and, for sign, i-ac-

cording as the corresponding arc is situated on the exterior
(as Ph+ZPh)PhPh+1in Fig. 1) or on the interior (as Ph+lPh+Z)
of the rectilinear triangle PhPh+lPh+2. T h e acute angle, a t
Ph+l or a t Ph+2, comprised between the arc aPh+lPh+) nd t h e

Small Curvilinear Triangles 211

chord P h + l P h + Z , is equal t o half the central angle of magni-

tude Zh/Rh.It is convenient t o give a sign t o this angle by
denoting it Ph, with

(1.1) 2Oh lhYh, h=l, 2, 3.

T h e absolute value l@hl is thus precisely the acute angle
between the arc and its chord; and, according t o the defini-

tion of Y h , the sign of P h is k according as the arc p h + l P h + Z

is situated on the exterior or on the interior (as in Fig. 1)
of the rectilinear triangle of chords.

It is easy t o express the sides a h and the angles CYh of the

rectilinear triangle of chords, as functions of the elements
h , (Ph of the circular triangle and of the curvatures Y h . I n
putting these expressions in the elementary formula of recti-
linear trigonometry, we shall obtain from them all the rela-
tions between the lh, the (Ph, and the Y h , which we propose
t o establish; these relations, duly simplified according t o
the smallness of t h e arcs lh, relative t o the respective radii,
will be precisely the trigonometric relations for the circular
triangles.

So much for the general thesis. Passing t o the details,

we first establish expressions for the angles p h in terms of
the CY), and t he auxiliary Ph.

If, as for P h in Fig. 2 , the circular arcs are concave toward

the interior of the rectilinear triangle P h P h + l P h + Z , then @h+l
and @h+2 are both positive according t o ( l . l ) , and we have
immediately

This formula holds in every case. Indeed, if one of the
circular sides abutting on P h is, in the neighborhood of the
point, interior to the rectilinear triangle, the corresponding

212 Levi-Civita Lectures

+I

Fig. 2

8, is negative by (1.1)) and it is just this sign that we must
give t o the angle in the expression (1.2) for ph.

Since the chord ah subtends the arc Zh) we have

ah=2Rh sin(+lh/Rh).

When we replace I/& by + v h ) the ambiguity in sign dis-
appears in every case, and we retain

ah= -2 sin(iLyh), h = 1, 2 , 3.

Yh

Consider now any one of the formulz of ordinary trigo-
nometry. It is a relation between the a’s and the CY’S, and
can be rewritten by replacing ah by its value (1.3), and CYh

Small Curvilinear Triangles 213

by its value obtained from (1.2); that is t o say, according
to (l.l), by

If, and such is the case, we are content with first approxi-

mations with respect t o the products LhYh, we may neglect
in the course of our calculations all terms of the second and
higher orders relative to unity, and the results are very
simple.

We give explicitly, by way of example, the formula for
the sine function,

(1.5) h =1, 2, 3,

where 1, independent of h, designates a small quantity of

the order of the dimensions of the triangle; namely, 1 is the

length of the diameter of the circumscribed triangle.

First of all, neglecting terms of the second and higher

orders relative t o unity, we can identify the sine with the

arc, and we obtain as a consequence of (1.3) the result t h a t

we may write

+01,(1.6)
ah = { 1

where the symbols 0,0, designate a set of terms of

-order 1, 2, a t least, with respect t o our argument Z h ~ h .

In order to obtain our trigonometric formula to the same
order of approximation, we have only t o replace in (1.5) the
quantities a h and (Yh by their values (1.6) and (1.4).

Taking into account for the moment only the finite terms
in a h , we obtain from (1.6)

+ a}( 1 . 7 ) .
Zh =1{ sin (ph

214 Levi-Civita Lectures

Next, expanding sinah according t o (1.4), and taking into
account only terms of the first order, we have

- 0.sin a h =sin (Ph 3 ( l h + 1 Y h + l + Z h + Z Y h + Z ) C O S ~ h +

I n the terms of the first order we can replace l h + l and /h+p
by their values as given in (1.7)) and obtain

(1.8) sin ah=sin (Ph-+l(yh sinph+Yh+lsinph+l
+Yh+Z sin (Ph+Z)COS(Ph+glYh sin (Ph cos (Ph+ @ a

To abridge the notation, we introduce the triangular cur-

vature 7;that is to say, the trinomial

(1.9) 7=i(Yh sin (Ph+Yh+l sin (Ph+l+Yh+Z sin (Phi?).

We now obtain. from ( l S ) , (1.6), and (1.8) the formula9

(1.10) -=I(lh 1ffrl(yhCOS(Ph-37COt~h)f @] ,h= i,2,3,

sin (Ph

which we desired t o establish.
We see here a good example of the correction due t o the

curvature of the sides, which affects the relation of simple
proportionality holding for rectilinear triangles. I n (1.10)

figures the auxiliary quantity 2, which, according t o (l.S),

is the diameter of the circle circumscribed about the tri-
angle of chords; that is t o say, without mentioning the
chords, the circle passing through the three vertices Ph.

I n order to obtain a set of three independent relations

between the sides Zh and the angles qh of the curvilinear

triangle, it suffices t o obtain an expression for I , as function
of the h,) (Ph, distinct from (1.10) itself. We obtain this im-

mediately by taking the well known expression of 1 as func-
tion of the sides ah of the rectilinear triangle, and then re-
placing each a h by its value (1.3); this amounts substan-
tially, according to (1.6), neglecting again here terms of the
second order, t o writing in place of ah.

Small Curvilinear Triangles 215

For the triangle of chords, we have, for each h,

.I=--- a h = ahah+lah+Z
sin ah ah+lah+2 smolt,

T h e denominator is twice the area of the triangle; that is,

',- 12 I p ( p a (l> p-az) ( p-al>

where

(1.11) p =Ha1+a2+4

designates the semi-perimeter.
Introducing the ratios

aPh--ph, h = 1, 2, 3,

which, since each side is less than the sum of the other two,
are proper fractions, we can write

(1.12)

According t o (1.6), the definition (1.11) of p takes the

form

(1.13)

Thus, neglecting terms of the second order relative t o unity,

we see that p i s the semi-perimeter of the given curvilinear

triangle; similarly, according t o (1.6),

so that
(1.14)

=${l+@].

216 Levi-Civita Lectures

In short, the equation to associate with the theorem of
sines (l.lO), for plane curvilinear triangles, is (1.12), where
(as first approximation, the approximation adopted up to

now) we can calculate p and the Ph from the given triangle:
p is the semi-perimeter and P h = l h / p , so t h a t

(1.15) Pl+PFI+P3 =2.

It is perhaps worth while to observe that, as a third equa-

tion t o be associated with (l.lO), where l has the meaning
of an auxiliary, indeterminate constant, we may use

Cr1+CQ+a3=7T,

which is a goniometric identity for a rectilinear triangle,
but becomes, according t o (1.4), an effective relation be-
tween the elements of the curvilinear triangle under con-
sideration.

In an analogous manner, we can generalize the other
formula: of ordinary rectilinear trigonometry, t o the case

of triangles formed by three circular arcs. I just point out

this small old-fashioned task t o those who wish t o carry
the calculations t o the end.

2. LOCAL R E P R E S E N T A T I O N O F PLANE CURVES

Let a plane curve C be given, let Q be an arbitrary point

on C, let s be the curvilinear abscissa of Q measured from

a fixed point P of C, and let the curve be regular between
P and Q inclusive. If x, y designate Cartesian coordinates,

the functions

(2.1) x=x(s), Y = y W ,

which define the curve C in parametric form, shall be sup-

posed t o be continuous, together with all the derivatives

which we shall have occasion t o consider, along the arc PQ.

Small Curvilinear Triangles 217

I n the same sense, the point Q and the unit vectors t and n

are regular functions of s, where f is directed along t h e tan-
gent t o C a t Q, in the positive sense of s, and n is directed
along the normal, in such a sense that the couple t, n is
oriented as are the positive directions of the coordinate
axes x , y.

For every point Q, we have the classical formula

and the formulae of Frenet,

d-dsf-- 7%

(2.4) -dn-- -7t,

ds

+,where y denotes the curvature of C with sign, and precisely

with the sign if n is directed toward the concavity of C,
but with the sign - in the contrary case.6

Differentiating (2.3) with respect t o s, and replacing d n / d s
by its value (2.4), we obtain

(2.5) dds2zf-- - 7 2 1 +.in,

where the superposed dot is an abridged notation for d / d s .
Applying the Maclaurin development to the point Q(r),

we have

.(2.6) Q(s) = P f s t + + ~ * y n + + s ~ ( - ~ ~ t + . i n ) +#,

the terms not written being of order superior t o the third

with respect t o s. Of course, the vector and scalar coeffi-
cients are all calculated a t the point P.

Such a development can be extended by calculating the

218 Levi-Civita Lectures

successive derivatives of Q, and consequently of t, accord-
ing t o (2.3), and by eliminating each time the derivatives
of n, according t o (2.4) and its derivatives.

Formula (2.6) shows us that, in order t o carry the de-
velopment just t o include the term in 53, we must introduce
y and 9. We see immediately, in reasoning by induction,
that in carrying the development just t o include the term

in P, we must introduce the derivatives of y at P through

the order m-2.
We take account now of the essential circumstance that

we are dealing with small arcs. Generalizing what we ad-
mitted in the introduction regarding the product FL, where

r is the maximum of the curvature and L is the length of

the small arc considered, we admit that the successive terms
in the development of &(s) according to the powers of J,
which (a factor J being taken separately) are pure num-
bers, have an order of smallness not less than m-1 for

the term in sm. If this is granted, we can in particular make
formula (2.6) specific, as far as the order of approximation

is concerned, by writing

If we suppose the origin of coordinates to be a t the point
P,we obtain a canonical representation of a plane curve in

the neighborhood of one of its points.' We have only t o
take rectangular components in the above vectorial rela-
tion; that is t o say, we have only t o decompose the second
member according t o t and n. We obtain

(2.8) x =s { 1- p y 2 + 0),

sy+W+O),

where @ depends on the values of y, 9 on the entire arc

PQ.

Small Curvilinear Triangles 219

3. GEODESIC CURVATURE AND THE FORMULIE OF FRENET

FOR A CURVE DRAWN ON A SURFACE

On a surface u we can consider, as in the plane, a curve
C and a vector w,which is a function of the variable point

Q on C.
Let the surface be given in terms of any curvilinear co-

ordinates d , x2 whatsoever, let

(3.1)

be the expression of its linear element, and let wi be the

contravariant components of the vector w.
We designate here too by s the length of arc on C, meas-

ured from a fixed point P, so that

are parametric equations of the curve C. Then the unit

vector t, tangent to C a t the variable point Q of curvilinear

abscissa 5, has for contravariant components

(3.3)

This being granted, it is necessary again for us t o recall
that we can generalize in an invariant manner the notion
of the derivative of the vector w with respect t o s, the

length of the curve C, by defining this vector Dw by means

of the components'

(3.4)

which, as easily could be verified, have contravariant be-

220 Levi-Civita Lectures

havior, where the denote the Christoffel symbols of the
second kind for ds2.

It follows furthermore from (3.4) t h a t this same vector
Dw has for covariant components

(3.5)

We note in passing that, if w is defined not only on C, but

also on a two-dimensional domain of Q, in the neighborhood

of C, we can introduce the partial derivatives of wi with

respect t o X I and with respect t o x2; we have then

- 2dwi awi dxh
h -axh -9ds
ds

and this permits us to write the contravariant components
(3.4) of Dw in the form

or again, since the quantity in brackets is the covariant
derivative wfh,

froin this last formula we perceive the contravariant char-
acter of the components ( D w ) ~F. or the case actually con-

sidered, of a vector w defined uniquely on C, I refer t o my

book previously cited.

If in particular we make the vector w coincide with t,

we know that
D f = -1n ,
r

where n denotes the surface vector normal to f toward the

Small Curvilinear Triangles 221

concavity of C, and ;1 denotes the absolute value of the

geodesic curvature. If v denotes +n, and we put for con-

formity

y = f-1,
r

we obtain

(3.6) Dt =yv,

where y is the geodesic curvature of C, with sign, and pre-

+cisely with the sign if v is directed toward the concavity
-of C, the sign in the contrary case. The vector v has the

two properties of being a unit vector and of being perpen-
dicular t o f. I n accordance with the metric of the surface u,

these facts are translated into the two formulae

=o,2 2
vivi = 1, v"ti

11

v i and t i denoting, of course, covariant components.
Differentiating the first equation with respect t o 5, we

obtain

If we replace dvi/dr, according t o (3.4), by

and dvi/dJ, according t o ( 3 3 , by

we find after a reduction that

222 Levi-Civita Lectures

The two sums are equal, each of them representing the
scalar product of DV and V , in accordance with the metric
(3.1). This product is therefore null, whence we are as-
sured that the vector DV is perpendicular t o V , on 6, and
consequently has the same direction, if not the same sense,
as 1.

We can therefore write

(3.7) D v = -y * i ,

y* representing a scalar factor as yet undetermined. I n
order t o determine it, we have only to differentiate with
respect to I the relation of orthogonality

2

Civiti=0,

1

and then t o replace, as always, the derivatives dvi/ds and
dtilds by their values obtained from (3.4) and (3.5). We

have thus

=o.ki[(DV)%+ ( D t ) i Y i ]

1

But, in terms of covariant components, (3.6) is equivalent
to

( D t )i =yvi.

Similarly, for contravariant components, (3.7) gives

( D v ) i = -y*ti.

Since t and v are both of length 1, the equality arising from
differentiation becomes therefore

-y*+y =O.

The expression for DV acquires precisely the form

(3.5) D v = -yf,

Small Curvilinear Triangles 223

which is exactly the second formula of Frenet for curves
on u, the first being (3.6).'

Returning to the contravariant components (3.4),putting
w = t and writing ti in place of dxd/ds, we obtain the two
formulae of Frenet (3.6) and (3.8) explicitly in the form

(3.9)

(3.10) i =1,2.

For simplification, it is convenient t o refer t o isothermic

coordinates X I , x2; t h a t is, coordinates such t h a t ds2 reduces

to the form

(3.11) +dr2=X(dx12

It is convenient, moreover, t o concern ourselves especially

with the canonical formlo relative t o the point P, which

is the origin of the arc s. Then, in a neighborhood of P,

of either the first or the second order, everything behaves

as if X had the form 9
1

1+-(K4x12+x22)

where the coordinates XI, x2 vanish a t P , and where K de-

notes the total curvature of the surface u a t the point P.
If we adopt the obvious notation

we see a t once that, at the point P ,

224 Levi-Civita Lectures

- The values of the second derivatives can be written to-

gether in the formula

(3.13) -h i k = i, k =1,2,

where the 8ik are the symbols of Kronecker (0 for i ~ k ,

1 for i = k ) .
We have, consequently,

Since a i l is zero for l ~ ani d equal t o unity for I=i, we

have simply

Introducing these values in (3.9), replacing ti in the first
member by dxilds, and using the relation

1,X ( P + t 2 2 ) =

we obtain

(3.15)

A t the point P,we have simply

(3.16) i = l , 2.

P

Similarly, (3.10) reduces a t the point P to

(3.17)

Differentiating (3.15) with respect t o s, and taking account
of the fact that, for a function f(xl, x 2 ) such as A, X i , or x i k ,

Small Curvilinear Triangles 225

where 3) denotes (dy/ds)p, and where the omitted terms

vanish a t P.

Taking account of formulae (3.13)) we have

($')(3.18) -=-jvi -pti+ Kti -+Kti=y V i + ( + K - y2)ti.

P

At the point P, we have X = l ; and the components

t1 =dxl/ds, tz=dxZ/ds of an arbitrary direction f coincide,

according t o (3.11), with their direction cosines, that is t o

say, with cos e, sin e, where e is the angle which the vector

t makes with the positive direction of the x1-line (x2=const.).

Similarly, v has components

since v is always turned through an angle w / 2 from t in the

positive sense of rotation, which is the sense of circulation

P h P h + I P h + Z when we are considering a triangle.

It follows from (3.3), (3.16), and (3.18), t h a t for the

point P, rg)p($)p=cos 9,

?r$$) $) rg)(3.19) =sin e,

= - y sin e, = y cos e,

P P

= ( + ~ - y 2 ) c o se-y sin e,

P

(ZJP e+?d3X2 = ( + ~ - y 2 )sin cos e.

226 Levi-Civita Lectures

We are finally able t o write, as in the preceding section, and
with the same remark concerning the arcs, which are re-
garded as being m a l l , the developments of the coordinates

X I , X* of a curve extending from the point P in the direction
e measured from the positive direction of the xl-line, in

terms of powers of the arc s; we give these explicitly t o
the third order inclusive. I n the first place, since XI and x2
vanish for s=O, we have

where the derivatives a t the point P are given by (3.19).

I n taking into account the factors cos e, sin e, we can

write -e eX ~ = C O S yI+sin z1,
(3.20)
. .x2 =cos e y2+ sin e 22,

where

Evidently, the formulas (3.21) are the parametric ex-

pressions of the coordinates of a curve C tangent at P t o

the positive x1-line, where e=O, while those of (3.22) refer
t o a curve tangent t o the orthogonal direction x2.

P.4. GEODESIC CURVE ISSUING FROM

P.ANGLE WITH T H E CURVE AT T H E POINT LENGTH

If we are concerned in particular with a geodesic, so that

y=?=O, we obtain (3.21) and (3.22) in the form

Small Curvilinear Triangles 227

a},y’ = 5 { 1+&K52+ yz =5@;
+ A m +{21 =5@, 22 = 5 1
0).

The parametric representation (3.20) reduces therefore to

eX ~ = C O S . s(I+&KP)+J@,

-(4.1) x2=sin e s ( l + & K ~ ) + s @ .

Consider on C a point PI,of which the curvilinear abscissa

is I , and suppose that l > O ; that is, suppose that arcs are

considered positive when measured from P toward P’. For

simplicity, suppose we have chosen a t P the direction of

the positive &line t o be tangent t o C. Then the parametric

representation (3.21) applies t o C, and we have only t o

set s=Z t o obtain the coordinates,

o},1I 1+&K-r2)12+&rl++VlZ+ @},

of the point P‘.
The same point PI belongs moreover also to the chord

PP’,represented parametrically by (4.1) and corresponding

t o s=u. Equating and dividing by I, we obtain

which define a l l and e as functions of 1 and of the quan-

tities y, y relative t o the curve C, and K relative t o the

surface u, all evaluated a t the point P.

Immediately, dividing the second equation by the first,
we obtain

showing, as was u priori evident, that tan e is a small quan-

228 Levi-Civita Lectures

tity of the first order. Moreover, by the Maclaurin de-

velopment, we have tan 6 = e + @, since tan e is an odd func-

tion, and therefore

e =+yl+$--p+ @,

(4.3) COS e = 1 -QyP+ 0.

With the above value for cos 8, the first of equations (4.2)

gives successively, when we neglect terms of the first order,

-0.= 1 + 0 ,
I

then the second,

-UI= 1 + @ ,

and finally only the third,

I n the last binomial we have written I in place of a, since
the difference between KP and Ka2 is not merely of the
third, but actually of the fourth order. We have thus

+ +a

1
= { 1 ++(+IC- y2)Pj (1 iy212)(1 -&KP) 0

= 1 -&-hy*P+ 0,

or, if we please,

(4.4) a =I { 1-&#w+a],

a well-known relation between the length of a small arc
and its chord, for a plane curve or for a curve of ordinary
space." For a curve drawn on an arbitrary surface, where
y denotes its geodesic curvature and a is equal to the length

Small Curvilinear Triangles 229

of the geodesic chord, the result has already been noted by
Darboux,12 who has carried out the calculation through the
fourth order inclusive. Even in stopping at the third order
we could expect K t o be involved, since in the course of
the calculation we are concerned with terms of the third
order in the parametric development, which depend in gen-
eral on the first and second derivatives of the coefficients

of ds2. The result shows that it is not so; nevertheless, I

have not succeeded in accounting for this by any synthetic
considerations.

Remark. I n formula (4.4),the value of y corresponds to
the point P , one of the two extremities common t o the arc

and t o its chord. T o the same order of approximation, we
can replace the value of y a t P by its value a t the other

extremity P’, or more generally a t an arbitrary point M
interior t o the arc PP’. Indeed, if sgZ is the curvilinear
abscissa P M of M , we have, according t o the formula of

finite increments,

YM=Ytl+Ol;

this assures the validity of (4.4),out t o terms of the second
order inclusive, for an arbitrary point M of the arc PP’ a t

which we calculate the value of y.

5 . RETURN T O T H E TRIANGLE PhPh+lPh+z.

TRIANGLE OF GEODESIC CHORDS. RELATIONS BETWEEN
THE SIDES AND THE ANGLES OF THESE TWO TRIANGLES

We return t o the triangle PhPh+lPh+2t o remark that, from
the qualitative point of view with respect t o sense and sign,
the entire development proceeds, if the triangle is suffi-
ciently small, as in the case of small circular triangles in
the plane ($1).

230 Levi-Civita Lectures

Henceforth, we fix our attention on an arbitrary vertex
P h , and continue t o denote by (oh the angle of the given
triangle, formed at P h by the two curvilinear arcs P h P h + l
and P h P h + 2 , by (Yh the angle formed a t P h by the correspond-
ing geodesic chords P h P h + 1 and P h P h + 2 . As in $1, we find

it convenient to introduce the two angles p comprised at

the point P h between the arcs P h P h + l , P h P h + Z and their
chords. We designate them p h ,h+l and p h ~ h+2, respectively,
the j’irst index referring to the vertex. I n $1, because of a

property of the circle, we had only three angles p, and the

first index sufficed. For example, with respect t o the arc
P h P h + i , We had t W O equal angles P h , h + l and b h + l , h , both Of
which we designated ph+2. Here we have introduced six p’s
with two distinct indices each. Their evaluation has been
substantially made in the preceding section : we have only
duly to specify formula (4.3).

We apply formula (4.3) first t o the angle p h , h + l com-
prised a t the point P h between the arc P h P h + l and its chord.

Then Ph,h+l identifies itself with e provided we replace 1 by

the length b + 2 of the arc P h P h + l opposite the vertex Ph+2,
and Y, 7 by Y h + % ?h+2, or more specifically by (Yh+2)Ph,
(?h+Z)P),, if we wish t o exhibit the point of the arc P h P h + l

t o which the value of the curvature and its derivative refer.
Thus we have

T h e sign of Ph,h+l is that of the preponderant term; that
is t o say, since the lengths 1 are essentially positive, the sign
of ( Y ~ + ~T)h~is~fo. llows necessarily in conformity with re-

marks made in $1 relative t o formula (1.1). Of course, we
can give t o h the values 1, 2, 3, with our usual convention
of reducing the indices h + l , h+2 modulo 3. The notation

indicates differentiation with respect t o the length of the

Small Curvilinear Triangles 231

arc P h P h + l , taken from P h toward P h + l , t h a t is, in the sense
PhPh+lPh+2 adopted as positive for tracing the perimeter of
the triangle.

Consider now the other arc P h P h + 2 abutting on the point
Ph, and the angle ph,h+2 which it forms with the correspond-
ing chord PhPh+2. I t s expression, furnished directly by (4.3),
is

where the index h+l refers t o the arc P h P h + 2 , opposite the
vertex Ph+l: here evidently we must regard the derivative
*jh+l of the curvature as being evaluated, with respect t o
the arc P h P h + 2 , for the direction toward Ph+2. The sense
does not coincide this time with the positive sense of circu-
lation PhPh+IPh+2, but is reversed. If w e m a k e the conven-
tion that the superposed dot always designates a deriva-
tive evaluated in the sense of circulation P h P h + l P h + S , adopted

+as positive o n the perimeter of the triangle, we must replace

the sign by the sign - in the second member of the above

formula. We then obtain

The rule t o remember is t h a t in the expressions of the p’s
of two indices, if the order of the indices is progressive, as

+,in formula (5.1), the second term has the sign while if

the order of the indices is retrogressive, as in formula (5.2),

the second term has the sign -.

I n formulae (5.1) and (5.2) appear y and ? relative t o
one same arc, namely, for a convenient choice of indices,
Ph+lPh+2, related one time t o P h + l ) the other time t o Ph+2.
To verify t h a t this is so, we write h + l in place of h in for-

232 Levi-Civita Lectures

mula (5.1), and take account of the point a t which the

evaluation is made, obtaining

&P h + l ,h+2 = (Yk)Ph+llh f + ( ? h ) +0;

and h+2 in place of h in (5.2), obtaining

Ph+P,h+l =&(yh)Ph+21h-+(?h)P~+lli+ 0.

We find it advantageous t o make our evaluations, for these

two formulae, a t the midpoint Mh of the arc P h + l P h + 2 . We
have, for the positive sense of circulation,

0,( 7 h ) M A l h = ( Y h ) P ~ , ~ h + & ( ? h ) P ~ , l i +

(?'h)PA+$h = ( " / h ) A I $ h f & ( ? h ) M h l i + 0.

Noting that

(?h)Ph+lli= (?h)M$i+ @ = ? h ( f 0,

where we have suppressed the specification as to point when
the evaluation is made a t M h , we obtain

- 0,2

( Y h ) Phillh =yhlh 3 Y h l h - k

f 0,( y h ) PA+Jh =r h l h $?hli+

and the preceding expressions for Ph+l,h+2 and Ph+O,h+l take
the definitive forms

where it is always the first index which denotes the vertex
t o which the angle is related; and Y h and ?h denote their
determination a t the midpoint M h of the arc P h + l P h + 2 .

There are formulae analogous t o (1.2) expressing the angle
@, of the given triangle by means of the corresponding angle

Small Curvilinear Triangles 233

(Yh of the triangle of chords and by means of the 0’s. They

are precisely

(oh = ah+ P h h+l+ P h , h+2;

when we solve for LYh) we have, according t o (5.3) and (5.4),

(5.5) -ah =(ph * ( Y h + l l h + l f Y h + 2 1 h + 2 ) +0,

- -1 ?h+21:+2) h = l , 2, 3.
n ( ? h + l /:+I

I n the relations ( 5 . 9 , as well as in the relations (4.3)
from which they were derived, the total curvature K of the

surface u in the neighborhood of the triangle does not figure
in terms up t o the second order inclusive. The same is
true of the expressions of the lengths of the geodesic chords
as functions of the curvilinear sides. Indeed, formula (4.4),

related t o the arc Ph+lPh+2 of length I,, gives, for the geo-

desic chord ah,

(5.6)

where it is permissible, according t o the last remark in $4)

t o regard Y h as being determined a t the midpoint Mh of
the arc Ph+IPh+%

We have in formula: ( 5 . 9 , (5.6) all that we shall need in
the next section in order t o establish, to second approxima-
tions, the trigonometry of small curvilinear triangles on u.

6. TRIGONOMETRIC RELATIONS

T h e triangle of geodesic chords has sides a h and angles (Yh.
I n our approximation, disregarding terms of the third and
higher orders, we can consider the surface u, in the neigh-
borhood of an arbitrary point 0 of u, as a variety of constant

curvature, K , the determination of K being computed a t

the point 0.13

It follows that, for the geodesic triangle having sides ah

234 Levi-Civita Lectures

and angles ah, the formula: of spherical or Lobatschewskian

trigonometry are valid, depending on the sign of K. We set

with

R>O for K>O,

and

-iR>O for K<O.

Confining ourselves t o the theorem of sines, we must trans-
form, either for the sphere or for the pseudosphere, the for-
mula:

[sinath[=-Rl I sin ah,
I(6.1) h = 1, 2, 3,

where for simplicity we have put

Now 1, independent of h, is called the modulus of the
spherical or pseudospherical triangle. Its value, expressed
for example by means of the sides alone, is given by

I=IRI(6.3) [sin a: sin u[ sin ail
11- c o s ~ a : - c o s ~ a ~ - c o s ~ a ~ + 2 c o s a : c o s a ~ c o s a : ) ~ '

Some interesting remarks might be made about the ex-

pression (6.3) of I, but for the sake of brevity we shall not

dwell on them. We restrict ourselves here again to develop-

ing the theorem of sines; that is t o say, we introduce in

formulz (6.1) the expressions (5.5) of the ffh, and the ex-

pressions (6.2), (5.6) of the &,, which may be written

Small Curvilinear Triangles 235

This is just what we need for our specified order of ap-
proximation. We have only to solve finally for the side lh
of the curvilinear triangle. One such calculation already

has been made, t o the first approximation, in 0 1 ; it led us
t o the expressions (1.10) of the lh as functions of the angles

(Ph (and of the yh and the auxiliary I). I n this connection,

it is important t o remark t h a t the auxiliary length I, intro-

duced actually as factor of proportionality in formulae (6.1),
is equal in the first approximation t o the quantity desig-
nated by the same letter in 01.

Indeed, starting from equations ( 6 . 1 ) and taking account
of the fact that a; is either real or pure imaginary, so that

is real in any case, we obtain immediately

Replacing [a’l by its value (6.4), and noting t h a t in virtue
of formulae (6.4),

we obtain

whence in particular

Ih=Isin (Yh(l+@j,
which coincides, up t o terms of the first order, with the
characterization of I which follows from ( 1 . 5 ) by means of
(1.6).

I n the parenthesis in the above formula for l h / [RI, when

we take account of the second order also, we can simply
replace Cyh by (Ph and likewise h by l s i n (Ph, since in doing

236 Levi-Civita Lectures

so we neglect only the third order relative t o unity. We

have therefore, writing K in place of 1/R2,

(6.5) l h = ~ s i nLYh(1+@ sin2 (Ph(K++-yi)+@].

There is left only the evaluation of sin a h , up t o the second
order inclusive, as function of the (Ph and 1.

Writing for the moment

(6.6) 8h =3(Yh+ilh+i+Yh+2lh+2) 2 +-ih+21;12+2),

+&(-ih+iZh+i

we obtain ( 5 . 5 ) in the form

sin LYh=sin (Ph cos tih-cos (Ph sin 8h+@.

Taking account of the circumstance that 8h is of the first
order, and neglecting as always the third order relative to
finite terms, we obtain

-sin ah=sin (Ph(l-$8,')-cos (Ph ah+@.

I n 8;. according t o (6.6), we may retain only the square of
the first term. There remains therefore

(6.7) sin ffh=sin (Ph{l -~(~h+llh+l+yh+21h+2)2)
-cos (Ph ' a h + @ ,

where 8h is defined by (6.6). The terms of orders 0 and 1
already have been calculated in $1, formula (1.8). Accord-
ingly, they can be written, by (1.9), sin (Ph and $h sin (Ph,
where

(6.8) $h=$l(Yh cos ( P h - 3 7 Cot (Ph).

On the other hand, the term Xh sin (Ph of second order in
(6.7) is, by (6.6),

Small Curvilinear Triangles 237

-sin (Ph (yh+llh+l +Th+Zlh+Z)

2

-cos (Ph ' ~ ( ? h + 1 1 ~ + 1 + 3 ' h + Z ~ h + Z ) .

Here we may replace &+I and h i 2 by 1 sin (Phil and 1 sin (Ph+Z,
and obtain

- I 2xh = sinf i ( Y h + l (Ph+l+Yh+Z sin (Ph+Z)'

Cot (Ph(?h+l sin2(Ph+l+?h+Z sin2(Ph+Z)} *

For symmetry, we introduce as in $1 the triangular curva-

ture

(6.9) 3

T = i x h y h sin (Phc

1

and the analogous trinomial

(6.10)

Now we can give t o the expression of xt,the form

(6.11) %= -1'{+(3T-Yh sin (Ph)2
+ACot ( P h ( 3 T ' - ? h sin2( P h ) } m

We always can write (6.7), when we separate terms of dif-
ferent orders, in the form

(6.12) .sin ah=sin (Ph {l++h+%+@}.

Noting t h a t sin Qh may be replaced by sin (Ph when it is
multiplied by a term of the second order, we obtain finally
from (6.5) and (6.12)

This is the form of the theorem of sines to the second ap-
proximation, with +h and Xh having values (6.8) and (6.11).

238 Levi-Civita Lectures

7 . REMARK ON THE THEOREM OF LEGENDRE

I n the calculations of the preceding section we have used,
for the triangle of chords, the formulae of spherical or pseudo-
spherical trigonometry. One might perhaps be inclined to
think that, in the approximation of the second order with
which we have contented ourselves, one could apply first
the theorem of Legendre, which, t o the same degree of ap-
proximation, reduces the resolution of a geodesic triangle
on a surface of constant curvature t o that of a plane tri-
angle having the same sides, and angles modified by one-
third of the excess ( c Y ~ + c Y ~ +AcctYual~ly), -th-i~s .is not
the case; and here is the reason.

One arrives a t the above enunciation of Legendre’s theo-
rem by establishing the fact that the same three inde-
pendent relations are satisfied up to the second order
inclusive by the elements of a spherical or pseudospherical

triangle and those of the corresponding plane triangle. It

does not follow in an absolute manner, since we are not
dealing with rigorous equalities, that every combination of
three relations is equally valid t o the same order of ap-
proximation. Precisely in the theorem of sines we see that
the equivalence subsists only in the first, but not in the
second approximation.

T h a t is why we have had t o take as point of departure
the formulae (6.1) of spherical or pseudospherical trigonom-
etry, without first reducing them by means of the theorem
of Legendre.

8. GENERAL INDICATIONS CONCERNING

APPROXIMATIONS OF HIGHER ORDER

I. The plane case. The parametric representation of

plane curves, considered in $2, can be extended, with the

Small Curvilinear Triangles 239

aid of the Taylor development according t o increasing
powers of the arc s, up t o an arbitrary order m. We make
the hypothesis, of course, that each arc of curve in question
behaves, with respect t o the geodesic curvature y and its
derivatives, in a manner analogous t o that admitted for
the first two approximations. We suppose in particular

that if I is the arc length and

y, 1,. . . ,$"-'I, y ( 4

the values of the curvature and its derivatives a t an ar-
bitrary point of the arc, then the products

. . .ly, 129, ,Im y(m-0, lm+'y(m),

which are pure numbers, are of order not less than

1, 2, * ' ,m, m+l,

respectively. Then, if we consider first the unit vector f,

tangent t o the curve, we see, from the formule of Frenet,

that t(s) can be represented, up t o the order m inclusive,

in the form +f (5)=A (s) lo B (s) no.
(8.1)

Here A(s) and B ( s ) are polynomials of degree ?n in J, the

coefficients being polynomials in the values of y and its

derivatives, 1, . , y(m-'), evaluated for s=O, or just as

well for any other value of s between 0 and I ; t o designates
the value of t for s=O, and no is fo turned through an angle
1r/2 in the positive sense.

If Q(s) designates the variable point of the arc under

consideration, we have

so that, by integration of f(s) from r=O, corresponding,

240 Levi-Civita Lectures

say, t o the point P,up t o a generic s, we obtain the para-

metric representation of the arc. For brevity we put

%(J) = f l d(J)dJ, 9 ( 5 ) = A S B( 5 )ds,
the y’s figuring in the preceding A and B being treated as

constants relative t o the integration, and obtain for the

other extremity P’ of the arc, corresponding t o s = I , Q =P’,

while, for the tangential versor (unit vector) t’ a t the point

P’, we set s = l in (8.1) and obtain

(5.3) t ’ = A (I)ro+B (1)no.

On the other hand, if u designates the versor of PI-P

and a its length, we have

P’-P=au,

so t h a t we can give t o the vectorial equation (8.2) the form

Now (8.4) is equivalent t o two scalar equations, while (8.3),
expressing the equality of two versors, yields only one. The
set of three equations allows us t o deduce, for the lunar

figure formed by an arc PP‘ and its chord PP’,both the

length a of the chord and also the values of the two angles

between chord and arc a t vertices P, P‘, as functions of

,I , y , y , . y* (m-1).

If now we consider, in the plane, a small triangle p h p h + l

p h + 2 having arbitrary curves as sides, we accordingly can
obtain for each side the length ah of the chord and the two

,w ( ~ - ’ ) .angles b + l , h + Z , Ph+Z,h+l as functions of lhy Y h , yhh, .

Small Curvilinear Triangles 241

And we have, as in 94,for the angles (Yh of the triangle of

chords,

- f ,a h =(ph ( P h + l , h+2 P h + 2 , h +l>

the (bh being the angles of the curvilinear triangle.

Thus we have only t o replace, in an arbitrary relation

of rectilinear trigonometry, the a h and the CYh as functions
- ,.of lh, (ph, y h , T h ,
yh(m-l), in order t o deduce the correspond-

ing formula, relative t o the curvilinear triangle, exactly up

to the order m inclusive.

11. The sphere. All the preceding can be carried over,

m u t a t i ~mutandis, t o arbitrary small curvilinear triangles

on a sphere or pseudosphere, proper account being taken,

in the course of the calculations and in the final formule,

of the value K of the constant total curvature.

111. Arbitrary surface. The method followed in 993-5,

with m=2, for an arbitrary surface, and the preceding sup-

plementary sketch, in the case of the plane, for an arbitrary

m, can be combined. W e accordingly introduce, not only

the derivatives of the geodesic curvature up t o the order

m-1, but also the value5 of the total curvature K and of

its partial derivative5 up to the order m-2, these last evalu-

ated a t an arbitrary point of the triangle.

It will be helpful, in connection with our general con-

siderations, t o recall a circumstance which is well known

for geodesic triangles.16 It is that we generally cannot at-

tain relations involving only sides, angles, geodesic curva-

ture y of the sides, total curvature K of the surface, and

their derivatives. This is because of the metric non-homo-

geneity of the surface u, which makes the position and orien-

tation of the triangle significant in the computations. I n

general, therefore, we cannot eliminate the coordinates of

the vertices, or any other absolute element which we might