# History of Topics in Special Relativity/Lorentz transformation (hyperbolic)

## Lorentz transformation via hyperbolic functions

### Translation in the hyperbolic plane

The case of a Lorentz transformation without spatial rotation is called a w:Lorentz boost. The simplest case can be given, for instance, by setting n=1 in the E:most general Lorentz transformation (1a):

{\displaystyle {\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}\\\hline {\begin{aligned}x_{0}^{\prime }&=x_{0}g_{00}+x_{1}g_{01}\\x_{1}^{\prime }&=x_{0}g_{10}+x_{1}g_{11}\\\\x_{0}&=x_{0}^{\prime }g_{00}-x_{1}^{\prime }g_{10}\\x_{1}&=-x_{0}^{\prime }g_{01}+x_{1}^{\prime }g_{11}\end{aligned}}\left|{\begin{aligned}g_{01}^{2}-g_{00}^{2}&=-1\\g_{11}^{2}-g_{10}^{2}&=1\\g_{01}g_{11}-g_{00}g_{10}&=0\\g_{10}^{2}-g_{00}^{2}&=-1\\g_{11}^{2}-g_{01}^{2}&=1\\g_{10}g_{11}-g_{00}g_{01}&=0\end{aligned}}\rightarrow {\begin{aligned}g_{00}^{2}&=g_{11}^{2}\\g_{01}^{2}&=g_{10}^{2}\end{aligned}}\right.\end{matrix}}}

or in matrix notation

{\displaystyle \left.{\begin{aligned}\mathbf {x} '&={\begin{bmatrix}g_{00}&g_{01}\\g_{10}&g_{11}\end{bmatrix}}\cdot \mathbf {x} \\\mathbf {x} &={\begin{bmatrix}g_{00}&-g_{10}\\-g_{01}&g_{11}\end{bmatrix}}\cdot \mathbf {x} '\end{aligned}}\quad \right|\quad \det {\begin{bmatrix}g_{00}&g_{01}\\g_{10}&g_{11}\end{bmatrix}}=1}

(3a)

which resembles precisely the relations of w:hyperbolic functions in terms of w:hyperbolic angle ${\displaystyle \eta }$. Thus a Lorentz boost or w:hyperbolic rotation (being the same as a rotation around an imaginary angle ${\displaystyle i\eta =\phi }$ in E:(2b) or a translation in the hyperbolic plane in terms of the hyperboloid model) is given by

{\displaystyle {\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}\\\hline g_{00}=g_{11}=\cosh \eta ,\ g_{01}=g_{10}=-\sinh \eta \\\hline \left.{\begin{aligned}&\quad \quad (A)&&\quad \quad (B)&&\quad \quad (C)\\x_{0}^{\prime }&=x_{0}\cosh \eta -x_{1}\sinh \eta &&={\frac {x_{0}-x_{1}\tanh \eta }{\sqrt {1-\tanh ^{2}\eta }}}&&={\frac {x_{0}-x_{1}v}{\sqrt {1-v^{2}}}}\\x_{1}^{\prime }&=-x_{0}\sinh \eta +x_{1}\cosh \eta &&={\frac {x_{1}-x_{0}\tanh \eta }{\sqrt {1-\tanh ^{2}\eta }}}&&={\frac {x_{1}-x_{0}v}{\sqrt {1-v^{2}}}}\\\\x_{0}&=x_{0}^{\prime }\cosh \eta +x_{1}^{\prime }\sinh \eta &&={\frac {x_{0}^{\prime }+x_{1}^{\prime }\tanh \eta }{\sqrt {1-\tanh ^{2}\eta }}}&&={\frac {x_{0}^{\prime }+x_{1}^{\prime }v}{\sqrt {1-v^{2}}}}\\x_{1}&=x_{0}^{\prime }\sinh \eta +x_{1}^{\prime }\cosh \eta &&={\frac {x_{1}^{\prime }+x_{0}^{\prime }\tanh \eta }{\sqrt {1-\tanh ^{2}\eta }}}&&={\frac {x_{1}^{\prime }+x_{0}^{\prime }v}{\sqrt {1-v^{2}}}}\end{aligned}}\right|{\scriptstyle {\begin{aligned}\sinh ^{2}\eta -\cosh ^{2}\eta &=-1&(a)\\\cosh ^{2}\eta -\sinh ^{2}\eta &=1&(b)\\{\frac {\sinh \eta }{\cosh \eta }}&=\tanh \eta =v&(c)\\{\frac {1}{\sqrt {1-\tanh ^{2}\eta }}}&=\cosh \eta &(d)\\{\frac {\tanh \eta }{\sqrt {1-\tanh ^{2}\eta }}}&=\sinh \eta &(e)\\{\frac {\tanh q\pm \tanh \eta }{1\pm \tanh q\tanh \eta }}&=\tanh \left(q\pm \eta \right)&(f)\end{aligned}}}\end{matrix}}}

or in matrix notation

{\displaystyle \left.{\begin{aligned}\mathbf {x} '&={\begin{bmatrix}\cosh \eta &-\sinh \eta \\-\sinh \eta &\cosh \eta \end{bmatrix}}\cdot \mathbf {x} \\\mathbf {x} &={\begin{bmatrix}\cosh \eta &\sinh \eta \\\sinh \eta &\cosh \eta \end{bmatrix}}\cdot \mathbf {x} '\end{aligned}}\quad \right|\quad \det {\begin{bmatrix}\cosh \eta &-\sinh \eta \\-\sinh \eta &\cosh \eta \end{bmatrix}}=1}

(3b)

Hyperbolic identities (a,b) on the right of (3b) were given by Riccati (1757), all identities (a,b,c,d,e,f) by Lambert (1768–1770). Lorentz transformations (3b-A) were given by Laisant (1874), Cox (1882), Goursat (1888), Lindemann (1890/91), Gérard (1892), Killing (1893, 1897/98), Whitehead (1897/98), Woods (1903/05), Elliott (1903) and Liebmann (1904/05) in terms of Weierstrass coordinates of the w:hyperboloid model, while transformations similar to (3b-C) have been used by Lipschitz (1885/86). In special relativity, hyperbolic functions were used by Frank (1909) and Varićak (1910).

Using the idendity ${\displaystyle \cosh \eta +\sinh \eta =e^{\eta }}$, Lorentz boost (3b) assumes a simple form by using w:squeeze mappings in analogy to Euler's formula in E:(2c):[1]

{\displaystyle {\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}\\\hline {\begin{matrix}{\begin{aligned}u'&=ku\\w'&={\frac {1}{k}}w\end{aligned}}&\Rightarrow &{\begin{aligned}x_{1}^{\prime }-x_{0}^{\prime }&=e^{\eta }\left(x_{1}-x_{0}\right)\\x_{1}^{\prime }+x_{0}^{\prime }&=e^{-\eta }\left(x_{1}+x_{0}\right)\end{aligned}}\quad {\begin{aligned}x_{1}-x_{0}&=e^{-\eta }\left(x_{1}^{\prime }-x_{0}^{\prime }\right)\\x_{1}+x_{0}&=e^{\eta }\left(x_{1}^{\prime }+x_{0}^{\prime }\right)\end{aligned}}\end{matrix}}\\\hline k=e^{\eta }=\cosh \eta +\sinh \eta ={\sqrt {\frac {1+\tanh \eta }{1-\tanh \eta }}}={\sqrt {\frac {1+v}{1-v}}}\end{matrix}}}

(3c)

Lorentz transformations (3c) for arbitrary k were given by many authors (see E:Lorentz transformations via squeeze mappings), while a form similar to ${\displaystyle k={\sqrt {\tfrac {1+v}{1-v}}}}$ was given by Lipschitz (1885/86), and the exponential form was implicitly used by Mercator (1668) and explicitly by Lindemann (1890/91), Elliott (1903), Herglotz (1909).

Rapidity can be composed of arbitrary many rapidities ${\displaystyle \eta _{1},\eta _{2}\dots }$ as per the w:angle sum laws of hyperbolic sines and cosines, so that one hyperbolic rotation can represent the sum of many other hyperbolic rotations, analogous to the relation between w:angle sum laws of circular trigonometry and spatial rotations. Alternatively, the hyperbolic angle sum laws themselves can be interpreted as Lorentz boosts, as demonstrated by using the parameterization of the w:unit hyperbola:

{\displaystyle {\begin{matrix}-x_{0}^{2}+x_{1}^{2}=-x_{0}^{\prime 2}+x_{1}^{\prime 2}=1\\\hline \left[\eta =\eta _{2}-\eta _{1}\right]\\{\begin{aligned}x_{0}^{\prime }&=\sinh \eta _{1}=\sinh \left(\eta _{2}-\eta \right)=\sinh \eta _{2}\cosh \eta -\cosh \eta _{2}\sinh \eta &&=x_{0}\cosh \eta -x_{1}\sinh \eta \\x_{1}^{\prime }&=\cosh \eta _{1}=\cosh \left(\eta _{2}-\eta \right)=-\sinh \eta _{2}\sinh \eta +\cosh \eta _{2}\cosh \eta &&=-x_{0}\sinh \eta +x_{1}\cosh \eta \\\\x_{0}&=\sinh \eta _{2}=\sinh \left(\eta _{1}+\eta \right)=\sinh \eta _{1}\cosh \eta +\cosh \eta _{1}\sinh \eta &&=x_{0}^{\prime }\cosh \eta +x_{1}^{\prime }\sinh \eta \\x_{1}&=\cosh \eta _{2}=\cosh \left(\eta _{1}+\eta \right)=\sinh \eta _{1}\sinh \eta +\cosh \eta _{1}\cosh \eta &&=x_{0}^{\prime }\sinh \eta +x_{1}^{\prime }\cosh \eta \end{aligned}}\end{matrix}}}

or in matrix notation

{\displaystyle {\scriptstyle {\begin{aligned}{\begin{bmatrix}x_{1}^{\prime }&x_{0}^{\prime }\\x_{0}^{\prime }&x_{1}^{\prime }\end{bmatrix}}&={\begin{bmatrix}\cosh \eta _{1}&\sinh \eta _{1}\\\sinh \eta _{1}&\cosh \eta _{1}\end{bmatrix}}={\begin{bmatrix}\cosh \left(\eta _{2}-\eta \right)&\sinh \left(\eta _{2}-\eta \right)\\\sinh \left(\eta _{2}-\eta \right)&\cosh \left(\eta _{2}-\eta \right)\end{bmatrix}}={\begin{bmatrix}\cosh \eta _{2}&\sinh \eta _{2}\\\sinh \eta _{2}&\cosh \eta _{2}\end{bmatrix}}\cdot {\begin{bmatrix}\cosh \eta &-\sinh \eta \\-\sinh \eta &\cosh \eta \end{bmatrix}}&&={\begin{bmatrix}x_{1}&x_{0}\\x_{0}&x_{1}\end{bmatrix}}\cdot {\begin{bmatrix}\cosh \eta &-\sinh \eta \\-\sinh \eta &\cosh \eta \end{bmatrix}}\\{\begin{bmatrix}x_{1}&x_{0}\\x_{0}&x_{1}\end{bmatrix}}&={\begin{bmatrix}\cosh \eta _{2}&\sinh \eta _{2}\\\sinh \eta _{2}&\cosh \eta _{2}\end{bmatrix}}={\begin{bmatrix}\cosh \left(\eta _{1}+\eta \right)&\sinh \left(\eta _{1}+\eta \right)\\\sinh \left(\eta _{1}+\eta \right)&\cosh \left(\eta _{1}+\eta \right)\end{bmatrix}}={\begin{bmatrix}\cosh \eta _{1}&\sinh \eta _{1}\\\sinh \eta _{1}&\cosh \eta _{1}\end{bmatrix}}\cdot {\begin{bmatrix}\cosh \eta &\sinh \eta \\\sinh \eta &\cosh \eta \end{bmatrix}}&&={\begin{bmatrix}x_{1}^{\prime }&x_{0}^{\prime }\\x_{0}^{\prime }&x_{1}^{\prime }\end{bmatrix}}\cdot {\begin{bmatrix}\cosh \eta &\sinh \eta \\\sinh \eta &\cosh \eta \end{bmatrix}}\end{aligned}}}}

or in exponential form as squeeze mapping analogous to (3c):

{\displaystyle {\begin{aligned}e^{-\eta _{1}}&=e^{\eta }e^{-\eta _{2}}=e^{\eta -\eta _{2}}&e^{-\eta _{2}}&=e^{-\eta }e^{-\eta _{1}}=e^{-\eta _{1}-\eta }\\e^{\eta _{1}}&=e^{-\eta }e^{\eta _{2}}=e^{\eta _{2}-\eta }&e^{\eta _{2}}&=e^{\eta }e^{\eta _{1}}=e^{\eta _{1}+\eta }\end{aligned}}}

(3d)

Hyperbolic angle sum laws were given by Riccati (1757) and Lambert (1768–1770) and many others, while matrix representations were given by Glaisher (1878) and Günther (1880/81).

### Hyperbolic law of cosines

By adding coordinates ${\displaystyle x_{2}^{\prime }=x_{2}}$ and ${\displaystyle x_{3}^{\prime }=x_{3}}$ in Lorentz transformation (3b) and interpreting ${\displaystyle x_{0},x_{1},x_{2},x_{3}}$ as w:homogeneous coordinates, the Lorentz transformation can be rewritten in line with equation E:(1b) by using coordinates ${\displaystyle [u_{1},\ u_{2},\ u_{3}]=\left[{\tfrac {x_{1}}{x_{0}}},\ {\tfrac {x_{2}}{x_{0}}},\ {\tfrac {x_{3}}{x_{0}}}\right]}$ defined by ${\displaystyle u_{1}^{2}+u_{2}^{2}+u_{3}^{2}\leq 1}$ inside the w:unit sphere as follows:

{\displaystyle {\begin{aligned}&\quad \quad (A)&&\quad \quad (B)&&\quad \quad (C)\\\hline \\u_{1}^{\prime }&={\frac {-\sinh \eta +u_{1}\cosh \eta }{\cosh \eta -u_{1}\sinh \eta }}&&={\frac {u_{1}-\tanh \eta }{1-u_{1}\tanh \eta }}&&={\frac {u_{1}-v}{1-u_{1}v}}\\u_{2}^{\prime }&={\frac {u_{2}}{\cosh \eta -u_{1}\sinh \eta }}&&={\frac {u_{2}{\sqrt {1-\tanh ^{2}\eta }}}{1-u_{1}\tanh \eta }}&&={\frac {u_{2}{\sqrt {1-v^{2}}}}{1-u_{1}v}}\\u_{3}^{\prime }&={\frac {u_{3}}{\cosh \eta -u_{1}\sinh \eta }}&&={\frac {u_{3}{\sqrt {1-\tanh ^{2}\eta }}}{1-u_{1}\tanh \eta }}&&={\frac {u_{3}{\sqrt {1-v^{2}}}}{1-u_{1}v}}\\\\\hline \\u_{1}&={\frac {\sinh \eta +u_{1}^{\prime }\cosh \eta }{\cosh \eta +u_{1}^{\prime }\sinh \eta }}&&={\frac {u_{1}^{\prime }+\tanh \eta }{1+u_{1}^{\prime }\tanh \eta }}&&={\frac {u_{1}^{\prime }+v}{1+u_{1}^{\prime }v}}\\u_{2}&={\frac {u_{2}^{\prime }}{\cosh \eta +u_{1}^{\prime }\sinh \eta }}&&={\frac {u_{2}^{\prime }{\sqrt {1-\tanh ^{2}\eta }}}{1+u_{1}^{\prime }\tanh \eta }}&&={\frac {u_{2}^{\prime }{\sqrt {1-v^{2}}}}{1+u_{1}^{\prime }v}}\\u_{3}&={\frac {u_{3}^{\prime }}{\cosh \eta +u_{1}^{\prime }\sinh \eta }}&&={\frac {u_{3}^{\prime }{\sqrt {1-\tanh ^{2}\eta }}}{1+u_{1}^{\prime }\tanh \eta }}&&={\frac {u_{3}^{\prime }{\sqrt {1-v^{2}}}}{1+u_{1}^{\prime }v}}\end{aligned}}}

(3e)

Transformations (A) were given by Escherich (1874), Goursat (1888), Killing (1898), and transformations (C) by Beltrami (1868), Schur (1885/86, 1900/02) in terms of Beltrami coordinates[2] of hyperbolic geometry. This transformation becomes equivalent to the w:hyperbolic law of cosines by restriction to coordinates of the ${\displaystyle \left[u_{1},u_{2}\right]}$-plane and ${\displaystyle \left[u'_{1},u'_{2}\right]}$-plane and defining their scalar products in terms of trigonometric and hyperbolic identities:[3][R 1][4]

{\displaystyle {\begin{matrix}&{\begin{matrix}u^{2}=u_{1}^{2}+u_{2}^{2}\\u'^{2}=u_{1}^{\prime 2}+u_{2}^{\prime 2}\end{matrix}}\left|{\begin{aligned}u_{1}=u\cos \alpha &={\frac {u'\cos \alpha '+v}{1+vu'\cos \alpha '}},&u_{1}^{\prime }=u'\cos \alpha '&={\frac {u\cos \alpha -v}{1-vu\cos \alpha }}\\u_{2}=u\sin \alpha &={\frac {u'\sin \alpha '{\sqrt {1-v^{2}}}}{1+vu'\cos \alpha '}},&u_{2}^{\prime }=u'\sin \alpha '&={\frac {u\sin \alpha {\sqrt {1-v^{2}}}}{1-vu\cos \alpha }}\\{\frac {u_{2}}{u_{1}}}=\tan \alpha &={\frac {u'\sin \alpha '{\sqrt {1-v^{2}}}}{u'\cos \alpha '+v}},&{\frac {u_{2}^{\prime }}{u_{1}^{\prime }}}=\tan \alpha '&={\frac {u\sin \alpha {\sqrt {1-v^{2}}}}{u\cos \alpha -v}}\end{aligned}}\right.\\\\\Rightarrow &u={\frac {\sqrt {v^{2}+u^{\prime 2}+2vu'\cos \alpha '-\left(vu'\sin \alpha '\right){}^{2}}}{1+vu'\cos \alpha '}},\quad u'={\frac {\sqrt {-v^{2}-u^{2}+2vu\cos \alpha +\left(vu\sin \alpha \right){}^{2}}}{1-vu\cos \alpha }}\\\Rightarrow &{\frac {1}{\sqrt {1-u^{\prime 2}}}}={\frac {1}{\sqrt {1-v^{2}}}}{\frac {1}{\sqrt {1-u^{2}}}}-{\frac {v}{\sqrt {1-v^{2}}}}{\frac {u}{\sqrt {1-u^{2}}}}\cos \alpha &(B)\\\Rightarrow &{\frac {1}{\sqrt {1-\tanh ^{2}\xi }}}={\frac {1}{\sqrt {1-\tanh ^{2}\eta }}}{\frac {1}{\sqrt {1-\tanh ^{2}\zeta }}}-{\frac {\tanh \eta }{\sqrt {1-\tanh ^{2}\eta }}}{\frac {\tanh \zeta }{\sqrt {1-\tanh ^{2}\zeta }}}\cos \alpha \\\Rightarrow &\cosh \xi =\cosh \eta \cosh \zeta -\sinh \eta \sinh \zeta \cos \alpha &(A)\end{matrix}}}

(3f)

The hyperbolic law of cosines (A) was given by Taurinus (1826) and Lobachevsky (1829/30) and others, while variant (B) was given by Schur (1900/02). By further setting ${\displaystyle \tanh \xi =\tanh \zeta =1}$ or ${\displaystyle u'=u=1}$ it follows:

${\displaystyle {\begin{matrix}(A)&\ \cos \alpha ={\frac {\cos \alpha '+\tanh \eta }{1+\tanh \eta \cos \alpha '}};&\ \sin \alpha ={\frac {\sin \alpha '{\sqrt {1-\tanh ^{2}\eta }}}{1+\tanh \eta \cos \alpha '}};&\ \tan \alpha ={\frac {\sin \alpha '{\sqrt {1-\tanh ^{2}\eta }}}{\cos \alpha '+\tanh \eta }};&\ \tan {\frac {\alpha }{2}}={\sqrt {\frac {1-\tanh \eta }{1+\tanh \eta }}}\tan {\frac {\alpha '}{2}}\\&\ \cos \alpha '={\frac {\cos \alpha -\tanh \eta }{1-\tanh \eta \cos \alpha }};&\ \sin \alpha '={\frac {\sin \alpha {\sqrt {1-\tanh ^{2}\eta }}}{1-\tanh \eta \cos \alpha }};&\ \tan \alpha '={\frac {\sin \alpha {\sqrt {1-\tanh ^{2}\eta }}}{\cos \alpha -\tanh \eta }};&\ \tan {\frac {\alpha '}{2}}={\sqrt {\frac {1+\tanh \eta }{1-\tanh \eta }}}\tan {\frac {\alpha }{2}}\\\\(B)&\ \cos \alpha ={\frac {\cos \alpha '+v}{1+v\cos \alpha '}};&\ \sin \alpha ={\frac {\sin \alpha '{\sqrt {1-v^{2}}}}{1+v\cos \alpha '}};&\ \tan \alpha ={\frac {\sin \alpha '{\sqrt {1-v^{2}}}}{\cos \alpha '+v}};&\ \tan {\frac {\alpha }{2}}={\sqrt {\frac {1-v}{1+v}}}\tan {\frac {\alpha '}{2}}\\&\ \cos \alpha '={\frac {\cos \alpha -v}{1-v\cos \alpha }};&\ \sin \alpha '={\frac {\sin \alpha {\sqrt {1-v^{2}}}}{1-v\cos \alpha }};&\ \tan \alpha '={\frac {\sin \alpha {\sqrt {1-v^{2}}}}{\cos \alpha -v}};&\ \tan {\frac {\alpha '}{2}}={\sqrt {\frac {1+v}{1-v}}}\tan {\frac {\alpha }{2}}\end{matrix}}}$

(3g)

Formulas (3g-B) are the equations of an w:ellipse of eccentricity v, w:eccentric anomaly α' and w:true anomaly α, first geometrically formulated by Kepler (1609) and explicitly written down by Euler (1735, 1748), Lagrange (1770) and many others in relation to planetary motions. They were also used by E:Darboux (1873) as a sphere transformation. In special relativity these formulas describe the aberration of light, see E:velocity addition and aberration.

## Historical notation

### Mercator (1668) – hyperbolic relations

While deriving the w:Mercator series, w:Nicholas Mercator (1668) demonstrated the following relations on a rectangular hyperbola:[M 1]

${\displaystyle {\begin{matrix}AD=1+a,\ DF={\sqrt {2a+aa}}\\AH={\frac {1+a+{\sqrt {2a+aa}}}{\sqrt {2}}},\ FH={\frac {1+a-{\sqrt {2a+aa}}}{\sqrt {2}}}\\AI=BI={\frac {1}{\sqrt {2}}}\\1+a=c,\ {\sqrt {2a+aa}}=d,\ 1=cc-dd\\AH*FH={\frac {cc-dd}{{\sqrt {2}}*{\sqrt {2}}}}={\frac {1}{2}}\\AI*BI={\frac {1}{2}}\\\hline AH*FH=AI*BI\\AH.AI::BI.FH\end{matrix}}}$

It can be seen that Mercator's relations ${\displaystyle 1+a=c}$, ${\displaystyle {\sqrt {2a+a^{2}}}=d}$ with ${\displaystyle c^{2}-d^{2}=1}$ implicitly correspond to hyperbolic functions ${\displaystyle c=\cosh \eta }$, ${\displaystyle d=\sinh \eta }$ with ${\displaystyle \cosh ^{2}\eta -\sinh ^{2}\eta =1}$ (which were explicitly introduced by Riccati (1757) much later). In particular, his result AH.AI::BI.FH, denoting that the ratio between AH and AI is equal to the ratio between BI and FH or ${\displaystyle {\tfrac {AH}{AI}}={\tfrac {BI}{FH}}}$ in modern notation, corresponds to squeeze mapping or Lorentz boost (3c) because:

${\displaystyle {\frac {AH}{AI}}={\frac {BI}{FH}}=1+a+{\sqrt {2a+a^{2}}}=c+d=\cosh \eta +\sinh \eta =e^{\eta }}$

or solved for AH and FH:

${\displaystyle AH=e^{\eta }AI}$ and ${\displaystyle FH=e^{-\eta }BI}$.

Furthermore, transforming Mercator's asymptotic coordinates ${\displaystyle AH={\tfrac {c+d}{\sqrt {2}}}}$, ${\displaystyle FH={\tfrac {c-d}{\sqrt {2}}}}$ into Cartesian coordinates ${\displaystyle x_{0},x_{1}}$ gives:

${\displaystyle x_{1}={\tfrac {AH+FH}{\sqrt {2}}}=c=\cosh \eta ,\quad x_{0}={\tfrac {AH-FH}{\sqrt {2}}}=d=\sinh \eta }$
which produces the unit hyperbola ${\displaystyle -x_{0}^{2}+x_{1}^{2}=1}$ as in (3d), in agreement with Mercator's result AH·FH=1/2 when the hyperbola is referred to its asymptotes.

### Euler (1735) – True and eccentric anomaly

w:Johannes Kepler (1609) geometrically formulated w:Kepler's equation and the relations between the w:mean anomaly, w:true anomaly, and w:eccentric anomaly.[M 2][5] The relation between the true anomaly z and the eccentric anomaly P was algebraically expressed by w:Leonhard Euler (1735/40) as follows:[M 3]

${\displaystyle \cos z={\frac {\cos P+v}{1+v\cos P}},\ \cos P={\frac {\cos z-v}{1-v\cos z}},\ \int P={\frac {\int z{\sqrt {1-v^{2}}}}{1-v\cos z}}}$

and in 1748:[M 4]

${\displaystyle \cos z={\frac {n+\cos y}{1+n\cos y}},\ \sin z={\frac {\sin y{\sqrt {1-n^{2}}}}{1+n\cos y}},\ \tan z={\frac {\sin y{\sqrt {1-n^{2}}}}{n+\cos y}}}$

while w:Joseph-Louis Lagrange (1770/71) expressed them as follows[M 5]

${\displaystyle \sin u={\frac {m\sin x}{1+n\cos x}},\ \cos u={\frac {n+\cos x}{1+n\cos x}},\ \operatorname {tang} {\frac {1}{2}}u={\frac {m}{1+n}}\operatorname {tang} {\frac {1}{2}}x,\ \left(m^{2}=1-n^{2}\right)}$
These relations resemble formulas (3g), while (3e) follows by setting ${\displaystyle [\cos z,\sin z,\cos y,\sin y]=\left[u_{1},u_{2},u'_{1},u'_{2}\right]}$ in Euler's formulas or ${\displaystyle [\cos u,\sin u,\cos x,\sin x]=\left[u_{1},u_{2},u'_{1},u'_{2}\right]}$ in Lagrange's formulas.

### Riccati (1757) – hyperbolic addition

w:Vincenzo Riccati (1757) introduced hyperbolic functions cosh and sinh, which he denoted as Ch. and Sh. related by ${\displaystyle Ch.^{2}-Sh.^{2}=r^{2}}$ with r being set to unity in modern publications, and formulated the addition laws of hyperbolic sine and cosine:[M 6][M 7]

${\displaystyle {\begin{matrix}CA=r,\ CB=Ch.\varphi ,\ BE=Sh.\varphi ,\ CD=Ch.\pi ,\ DF=Sh.\pi \\CM=Ch.{\overline {\varphi +\pi }},\ MN=Sh.{\overline {\varphi +\pi }}\\CK={\frac {r}{\sqrt {2}}},\ CG={\frac {Ch.\varphi +Sh.\varphi }{\sqrt {2}}},\ CH={\frac {Ch.\pi +Sh.\pi }{\sqrt {2}}},\ CP={\frac {Ch.{\overline {\varphi +\pi }}+Sh.{\overline {\varphi +\pi }}}{\sqrt {2}}}\\CK:CG::CH:CP\\\left[Ch.^{2}-Sh.^{2}=rr\right]\\\hline Ch.{\overline {\varphi +\pi }}={\frac {Ch.\varphi \,Ch.\pi +Sh.\varphi \,Sh.\pi }{r}}\\Sh.{\overline {\varphi +\pi }}={\frac {Ch.\varphi \,Sh.\pi +Ch.\pi \,Sh.\varphi }{r}}\end{matrix}}}$

He furthermore showed that ${\displaystyle Ch.{\overline {\varphi -\pi }}}$ and ${\displaystyle Sh.{\overline {\varphi -\pi }}}$ follow by setting ${\displaystyle Ch.\pi \Rightarrow Ch.-\pi }$ and ${\displaystyle Sh.\pi \Rightarrow Sh.-\pi }$ in the above formulas.

The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost (3d) with ${\displaystyle \pi =\eta ,\ \varphi =\eta _{1},\ {\overline {\varphi +\pi }}=\eta _{2}}$.

### Lambert (1768–1770) – hyperbolic addition

While Riccati (1757) discussed the hyperbolic sine and cosine, w:Johann Heinrich Lambert (read 1767, published 1768) introduced the expression tang φ or abbreviated as the w:tangens hyperbolicus ${\displaystyle {\scriptstyle {\frac {e^{u}-e^{-u}}{e^{u}+e^{-u}}}}}$ of a variable u, or in modern notation tφ=tanh(u):[M 8][6]

{\displaystyle \left.{\begin{aligned}\xi \xi -1&=\eta \eta &(a)\\1+\eta \eta &=\xi \xi &(b)\\{\frac {\eta }{\xi }}&=tang\ \phi =t\phi &(c)\\\xi &={\frac {1}{\sqrt {1-t\phi ^{2}}}}&(d)\\\eta &={\frac {t\phi }{\sqrt {1-t\phi ^{2}}}}&(e)\\t\phi ''&={\frac {t\phi +t\phi '}{1+t\phi \cdot t\phi '}}&(f)\\t\phi '&={\frac {t\phi ''-t\phi }{1-t\phi \cdot t\phi ''}}&(g)\end{aligned}}\right|{\begin{aligned}2u&=\log {\frac {1+t\phi }{1-t\phi }}\\\xi &={\frac {e^{u}+e^{-u}}{2}}\\\eta &={\frac {e^{u}-e^{-u}}{2}}\\t\phi &={\frac {e^{u}-e^{-u}}{e^{u}+e^{-u}}}\\e^{u}&=\xi +\eta \\e^{-u}&=\xi -\eta \end{aligned}}}

In (1770) he rewrote the addition law for the hyperbolic tangens (f) or (g) as:[M 9]

{\displaystyle {\begin{aligned}t(y+z)&=(ty+tz):(1+ty\cdot tz)&(f)\\t(y-z)&=(ty-tz):(1-ty\cdot tz)&(g)\end{aligned}}}
The hyperbolic relations (a,b,c,d,e,f) are equivalent to the hyperbolic relations on the right of (3b). Relations (f,g) can also be found in (3e). By setting tφ=v/c, formula (c) becomes the relative velocity between two frames, (d) the w:Lorentz factor, (e) the w:proper velocity, (f) or (g) becomes the Lorentz transformation of velocity (or relativistic w:velocity addition formula) for collinear velocities in E:(4a) and E:(4d).

Lambert also formulated the addition laws for the hyperbolic cosine and sine (Lambert's "cos" and "sin" actually mean "cosh" and "sinh"):

{\displaystyle {\begin{aligned}\sin(y+z)&=\sin y\cos z+\cos y\sin z\\\sin(y-z)&=\sin y\cos z-\cos y\sin z\\\cos(y+z)&=\cos y\cos z+\sin y\sin z\\\cos(y-z)&=\cos y\cos z-\sin y\sin z\end{aligned}}}
The angle sum laws for hyperbolic sine and cosine can be interpreted as hyperbolic rotations of points on a hyperbola, as in Lorentz boost (3d).

### Taurinus (1826) – Hyperbolic law of cosines

After the addition theorem for the tangens hyperbolicus was given by Lambert (1768), w:hyperbolic geometry was used by w:Franz Taurinus (1826), and later by w:Nikolai Lobachevsky (1829/30) and others, to formulate the w:hyperbolic law of cosines:[M 10][7][8]

${\displaystyle A=\operatorname {arccos} {\frac {\cos \left(\alpha {\sqrt {-1}}\right)-\cos \left(\beta {\sqrt {-1}}\right)\cos \left(\gamma {\sqrt {-1}}\right)}{\sin \left(\beta {\sqrt {-1}}\right)\sin \left(\gamma {\sqrt {-1}}\right)}}}$
When solved for ${\displaystyle \cos \left(\alpha {\sqrt {-1}}\right)}$ it corresponds to the Lorentz transformation in Beltrami coordinates (3f), and by defining the rapidities ${\displaystyle {\scriptstyle \left(\left[{\frac {U}{c}},\ {\frac {v}{c}},\ {\frac {u}{c}}\right]=\left[\tanh \alpha ,\ \tanh \beta ,\ \tanh \gamma \right]\right)}}$ it corresponds to the relativistic velocity addition formula E:(4e).

### Beltrami (1868) – Beltrami coordinates

w:Eugenio Beltrami (1868a) introduced coordinates of the w:Beltrami–Klein model of hyperbolic geometry, and formulated the corresponding transformations in terms of homographies:[M 11]

${\displaystyle {\begin{matrix}ds^{2}=R^{2}{\frac {\left(a^{2}+v^{2}\right)du^{2}-2uv\,du\,dv+\left(a^{2}+v^{2}\right)dv^{2}}{\left(a^{2}+u^{2}+v^{2}\right)^{2}}}\\u^{2}+v^{2}=a^{2}\\\hline u''={\frac {aa_{0}\left(u'-r_{0}\right)}{a^{2}-r_{0}u'}},\ v''={\frac {a_{0}w_{0}v'}{a^{2}-r_{0}u'}},\\\left(r_{0}={\sqrt {u_{0}^{2}+v_{0}^{2}}},\ w_{0}={\sqrt {a^{2}-r_{0}^{2}}}\right)\\\hline ds^{2}=R^{2}{\frac {\left(a^{2}-v^{2}\right)du^{2}+2uv\,du\,dv+\left(a^{2}-v^{2}\right)dv^{2}}{\left(a^{2}-u^{2}-v^{2}\right)^{2}}}\\(R=R{\sqrt {-1}},\ a=a{\sqrt {-1}})\end{matrix}}}$

(where the disk radius a and the w:radius of curvature R are real in spherical geometry, in hyperbolic geometry they are imaginary), and for arbitrary dimensions in (1868b)[M 12]

${\displaystyle {\begin{matrix}ds=R{\frac {\sqrt {dx^{2}+dx_{1}^{2}+dx_{2}^{2}+\cdots +dx_{n}^{2}}}{x}}\\x^{2}+x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=a^{2}\\\hline y_{1}={\frac {ab\left(x_{1}-a_{1}\right)}{a^{2}-a_{1}x_{1}}}\ {\text{or}}\ x_{1}={\frac {a\left(ay_{1}+a_{1}b\right)}{ab+a_{1}y_{1}}},\ x_{r}=\pm {\frac {ay_{r}{\sqrt {a^{2}-a_{1}^{2}}}}{ab+a_{1}y_{1}}}\ (r=2,3,\dots ,n)\\\hline ds=R{\frac {\sqrt {dx_{1}^{2}+dx_{2}^{2}+\cdots +dx_{n}^{2}-dx^{2}}}{x}}\\x^{2}=a^{2}+x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\\\left(R=R{\sqrt {-1}},\ x=x{\sqrt {-1}},\ a=a{\sqrt {-1}}\right)\end{matrix}}}$
Setting a=a0 Beltrami's (1868a) formulas become formulas (3e), or in his (1868b) formulas one sets a=b for arbitrary dimensions.

### Laisant (1874) – Equipollences

In his French translation of w:Giusto Bellavitis' principal work on w:equipollences, w:Charles-Ange Laisant (1874) added a chapter related to hyperbolas. The equipollence OM and its tangent MT of a hyperbola is defined by Laisant as[M 13]

(1) ${\displaystyle {\begin{matrix}&\mathrm {OM} \bumpeq x\mathrm {OA} +y\mathrm {OB} \\&\mathrm {MT} \bumpeq y\mathrm {OA} +x\mathrm {OB} \\&\left[x^{2}-y^{2}=1;\ x=\cosh t,\ y=\sinh t\right]\\\Rightarrow &\mathrm {OM} \bumpeq \cosh t\cdot \mathrm {OA} +\sinh t\cdot \mathrm {OB} \end{matrix}}}$

Here, OA and OB are conjugate semi-diameters of a hyperbola with OB being imaginary, both of which he related to two other conjugated semi-diameters OC and OD by the following transformation:

{\displaystyle {\begin{matrix}{\begin{aligned}\mathrm {OC} &\bumpeq c\mathrm {OA} +d\mathrm {OB} &\qquad &&\mathrm {OA} &\bumpeq c\mathrm {OC} -d\mathrm {OD} \\\mathrm {OD} &\bumpeq d\mathrm {OA} +c\mathrm {OB} &&&\mathrm {OB} &\bumpeq -d\mathrm {OC} +c\mathrm {OD} \end{aligned}}\\\left[c^{2}-d^{2}=1\right]\end{matrix}}}

producing the invariant relation

${\displaystyle (\mathrm {OC} )^{2}-(\mathrm {OD} )^{2}\bumpeq (\mathrm {OA} )^{2}-(\mathrm {OB} )^{2}}$.

Substituting into (1), he showed that OM retains its form

${\displaystyle {\begin{matrix}\mathrm {OM} \bumpeq (cx-dy)\mathrm {OC} +(cy-dx)\mathrm {OD} \\\left[(cx-dy)^{2}-(cy-dx)^{2}=1\right]\end{matrix}}}$

He also defined velocity and acceleration by differentiation of (1).

These relations are equivalent to several Lorentz boosts or hyperbolic rotations producing the invariant Lorentz interval in line with (3b).

### Escherich (1874) – Beltrami coordinates

w:Gustav von Escherich (1874) discussed the plane of constant negative curvature[9] based on the w:Beltrami–Klein model of hyperbolic geometry by Beltrami (1868). Similar to w:Christoph Gudermann (1830)[M 14] who introduced axial coordinates x=tan(a) and y=tan(b) in sphere geometry in order to perform coordinate transformations in the case of rotation and translation, Escherich used hyperbolic functions x=tanh(a/k) and y=tanh(b/k)[M 15] in order to give the corresponding coordinate transformations for the hyperbolic plane, which for the case of translation have the form:[M 16]

${\displaystyle x={\frac {\sinh {\frac {a}{k}}+x'\cosh {\frac {a}{k}}}{\cosh {\frac {a}{k}}+x'\sinh {\frac {a}{k}}}}}$ and ${\displaystyle y={\frac {y'}{\cosh {\frac {a}{k}}+x'\sinh {\frac {a}{k}}}}}$
This is equivalent to Lorentz transformation (3e), also equivalent to the relativistic velocity addition E:(4d) by setting ${\displaystyle {\tfrac {a}{k}}=\operatorname {atanh} {\tfrac {v}{c}}}$ and multiplying [x,y,x′,y′] by 1/c, and equivalent to Lorentz boost (3b) by setting ${\displaystyle \scriptstyle (x,\ y,\ x',\ y')=\left({\frac {x_{1}}{x_{0}}},\ {\frac {x_{2}}{x_{0}}},\ {\frac {x_{1}^{\prime }}{x_{0}^{\prime }}},\ {\frac {x_{2}^{\prime }}{x_{0}^{\prime }}}\right)}$. This is the relation between the Beltrami coordinates in terms of Gudermann-Escherich coordinates, and the Weierstrass coordinates of the w:hyperboloid model introduced by E:Killing (1878–1893), E:Poincaré (1881), and E:Cox (1881). Both coordinate systems were compared by Cox (1881).[M 17]

### Glaisher (1878) – hyperbolic addition

It was shown by w:James Whitbread Lee Glaisher (1878) that the hyperbolic addition laws can be expressed by matrix multiplication:[M 18]

${\displaystyle {\begin{matrix}{\begin{vmatrix}\cosh x,&\sinh x\\\sinh x,&\cosh x\end{vmatrix}}=1,\ {\begin{vmatrix}\cosh y,&\sinh y\\\sinh y,&\cosh y\end{vmatrix}}=1\\{\text{by multiplication:}}\\\Rightarrow {\begin{vmatrix}c_{1}c_{2}+s_{1}s_{2},&s_{1}c_{2}+c_{1}s_{2}\\c_{1}s_{2}+s_{1}c_{2},&s_{1}s_{2}+c_{1}c_{2}\end{vmatrix}}=1\\{\text{where}}\ \left[c_{1},c_{2},c_{3},c_{4}\right]=\left[\cosh x,\cosh y,\sinh x,\sinh y\right]\\\Rightarrow {\begin{vmatrix}\cosh(x+y),&\sinh(x+y)\\\sinh(x+y),&\cosh(x+y)\end{vmatrix}}=1\end{matrix}}}$
In this matrix representation, the analogy between the hyperbolic angle sum laws and the Lorentz boost becomes obvious: In particular, the matrix ${\displaystyle \scriptstyle {\begin{vmatrix}\cosh y,&\sinh y\\\sinh y,&\cosh y\end{vmatrix}}}$ producing the hyperbolic addition is analogous to matrix ${\displaystyle \scriptstyle {\begin{bmatrix}\cosh \eta &\sinh \eta \\\sinh \eta &\cosh \eta \end{bmatrix}}}$ producing Lorentz boost (3b) and (3d).

### Günther (1880/81) – hyperbolic addition

Following Glaisher (1878), w:Siegmund Günther (1880/81) expressed the hyperbolic addition laws by matrix multiplication:[M 19]

${\displaystyle {\begin{matrix}{\begin{vmatrix}{\mathfrak {Cos}}\,x,&{\mathfrak {Sin}}\,x\\{\mathfrak {Sin}}\,x,&{\mathfrak {Cos}}\,x\end{vmatrix}}\cdot {\begin{vmatrix}{\mathfrak {Cos}}\,y,&{\mathfrak {Sin}}\,y\\{\mathfrak {Sin}}\,y,&{\mathfrak {Cos}}\,y\end{vmatrix}}\\={\begin{vmatrix}{\mathfrak {Cos}}\,x\,{\mathfrak {Cos}}\,y+{\mathfrak {Sin}}\,x\,{\mathfrak {Sin}}\,y,&{\mathfrak {Cos}}\,x\,{\mathfrak {Sin}}\,y+{\mathfrak {Sin}}\,x\,{\mathfrak {Cos}}\,y\\{\mathfrak {Sin}}\,x\,{\mathfrak {Cos}}\,y+{\mathfrak {Cos}}\,x\,{\mathfrak {Sin}}\,y,&{\mathfrak {Sin}}\,x\,{\mathfrak {Sin}}\,y+{\mathfrak {Cos}}\,x\,{\mathfrak {Cos}}\,y\end{vmatrix}}\\={\begin{vmatrix}{\mathfrak {Cos}}\,(x+y),&{\mathfrak {Sin}}\,(x+y)\\{\mathfrak {Sin}}\,(x+y),&{\mathfrak {Cos}}\,(x+y)\end{vmatrix}}=1\end{matrix}}}$
In this matrix representation, the analogy between the hyperbolic angle sum laws and the Lorentz boost becomes obvious: In particular, the matrix ${\displaystyle \scriptstyle {\begin{vmatrix}{\mathfrak {Cos}}\,y,&{\mathfrak {Sin}}\,y\\{\mathfrak {Sin}}\,y,&{\mathfrak {Cos}}\,y\end{vmatrix}}}$ producing the hyperbolic addition is analogous to matrix ${\displaystyle \scriptstyle {\begin{bmatrix}\cosh \eta &\sinh \eta \\\sinh \eta &\cosh \eta \end{bmatrix}}}$ producing Lorentz boost (3b) and (3d).

### Cox (1881/82) – Weierstrass coordinates

w:Homersham Cox (1881/82) defined the case of translation in the hyperbolic plane with the y-axis remaining unchanged:[M 20]

{\displaystyle {\begin{aligned}X&=x\cosh p-z\sinh p\\Z&=-x\sinh p+z\cosh p\\\\x&=X\cosh p+Z\sinh p\\z&=X\sinh p+Z\cosh p\end{aligned}}}
This is equivalent to Lorentz boost (3b).

### Lipschitz (1885/86) – Quadratic forms

w:Rudolf Lipschitz (1885/86) discussed transformations leaving invariant the sum of squares

${\displaystyle x_{1}^{2}+x_{2}^{2}\dots +x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n}^{2}}$

which he rewrote as

${\displaystyle x_{1}^{2}-y_{1}^{2}+x_{2}^{2}-y_{2}^{2}+\dots +x_{n}^{2}-y_{n}^{2}=0}$.

This led to the problem of finding transformations leaving invariant the pairs ${\displaystyle x_{a}^{2}-y_{a}^{2}}$ (where a=1...n) for which he gave the following solution:[M 21]

{\displaystyle {\begin{matrix}x_{a}^{2}-y_{a}^{2}={\mathfrak {x}}_{a}^{2}-{\mathfrak {y}}_{a}^{2}\\\hline {\begin{aligned}x_{a}-y_{a}&=\left({\mathfrak {x}}_{a}-{\mathfrak {y}}_{a}\right)r_{a}\\x_{a}+y_{a}&=\left({\mathfrak {x}}_{a}+{\mathfrak {y}}_{a}\right){\frac {1}{r_{a}}}\end{aligned}}\quad (a)\\\hline {\begin{matrix}{\begin{aligned}2{\mathfrak {x}}_{a}&=\left(r_{a}+{\frac {1}{r_{a}}}\right)x_{a}+\left(r_{a}-{\frac {1}{r_{a}}}\right)y_{a}\\2{\mathfrak {y}}_{a}&=\left(r_{a}-{\frac {1}{r_{a}}}\right)x_{a}+\left(r_{a}+{\frac {1}{r_{a}}}\right)y_{a}\end{aligned}}\quad (b)\end{matrix}}\\\hline \left\{{\begin{matrix}r_{a}={\frac {\sqrt {s_{a}+1}}{\sqrt {s_{a}-1}}}\\s_{a}>1\end{matrix}}\right\}\Rightarrow {\begin{aligned}{\mathfrak {x}}_{a}&={\frac {s_{a}x_{a}+y_{a}}{{\sqrt {s_{a}-1}}{\sqrt {s_{a}+1}}}}\\{\mathfrak {y}}_{a}&={\frac {x_{a}+s_{a}y_{a}}{{\sqrt {s_{a}-1}}{\sqrt {s_{a}+1}}}}\end{aligned}}\quad (c)\end{matrix}}}
Lipschitz's transformations (c) and (a) are equivalent to Lorentz boosts (3b-C) and (3c) by the identity ${\displaystyle s_{a}={\tfrac {1}{v}}=\coth \eta }$. That is, by substituting ${\displaystyle v={\tfrac {1}{s_{a}}}}$ in (3b-C) or (3c) we obtain Lipschitz's transformations.

### Schur (1885/86, 1900/02) – Beltrami coordinates

w:Friedrich Schur (1885/86) discussed spaces of constant Riemann curvature, and by following Beltrami (1868) he used the transformation[M 22]

${\displaystyle x_{1}=R^{2}{\frac {y_{1}+a_{1}}{R^{2}+a_{1}y_{1}}},\ x_{2}=R{\sqrt {R^{2}-a_{1}^{2}}}{\frac {y_{2}}{R^{2}+a_{1}y_{1}}},\dots ,\ x_{n}=R{\sqrt {R^{2}-a_{1}^{2}}}{\frac {y_{n}}{R^{2}+a_{1}y_{1}}}}$
This is equivalent to Lorentz transformation (3e) and therefore also equivalent to the relativistic velocity addition E:(4d) in arbitrary dimensions by setting R=c as the speed of light and a1=v as relative velocity.

In (1900/02) he derived basic formulas of non-Eucliden geometry, including the case of translation for which he obtained the transformation similar to his previous one:[M 23]

${\displaystyle x'={\frac {x-a}{1-{\mathfrak {k}}ax}},\quad y'={\frac {y{\sqrt {1-{\mathfrak {k}}a^{2}}}}{1-{\mathfrak {k}}ax}}}$

where ${\displaystyle {\mathfrak {k}}}$ can have values >0, <0 or ∞.

This is equivalent to Lorentz transformation (3e) and therefore also equivalent to the relativistic velocity addition E:(4d) by setting a=v and ${\displaystyle {\mathfrak {k}}={\tfrac {1}{c^{2}}}}$.

He also defined the triangle[M 24]

${\displaystyle {\frac {1}{\sqrt {1-{\mathfrak {k}}c^{2}}}}={\frac {1}{\sqrt {1-{\mathfrak {k}}a^{2}}}}\cdot {\frac {1}{\sqrt {1-{\mathfrak {k}}b^{2}}}}-{\frac {a}{\sqrt {1-{\mathfrak {k}}a^{2}}}}\cdot {\frac {b}{\sqrt {1-{\mathfrak {k}}b^{2}}}}\cos \gamma }$
This is equivalent to the hyperbolic law of cosines and the relativistic velocity addition (3f, b) or E:(4e) by setting ${\displaystyle [{\mathfrak {k}},\ c,\ a,\ b]=\left[{\tfrac {1}{c^{2}}},\ {\sqrt {u_{x}^{\prime 2}+u_{y}^{\prime 2}}},\ v,\ {\sqrt {u_{x}^{2}+u_{y}^{2}}}\right]}$.

### Goursat (1887/88) – Minimal surfaces

w:Édouard Goursat defined real coordinates ${\displaystyle x,y}$ of minimal surface ${\displaystyle S}$ and imaginary coordinates ${\displaystyle x_{0},y_{0}}$ of the adjoint minimal surface ${\displaystyle S_{0}}$, so that another real minimal surface ${\displaystyle S_{1}}$ follows by the (conformal) transformation:[M 25]

{\displaystyle {\begin{aligned}x_{1}&={\frac {1+k^{2}}{2k}}x-{\frac {k^{2}-1}{2k}}y_{0}\\y_{1}&={\frac {1+k^{2}}{2k}}y+{\frac {k^{2}-1}{2k}}x_{0}\\z_{1}&=z\end{aligned}}}

and expressed these equations in terms of hyperbolic functions by setting ${\displaystyle k=e^{\varphi }}$:[M 26]

{\displaystyle {\begin{aligned}x_{1}&=x\cosh \varphi -y_{0}\sinh \varphi \\y_{1}&=y\cosh \varphi +x_{0}\sinh \varphi \\z_{1}&=z\end{aligned}}}
This becomes Lorentz boost (3b) by replacing the imaginary coordinates ${\displaystyle x_{0},y_{0}}$ by real coordinates defined as ${\displaystyle [x_{0},y_{0}]=[-x,y]}$. It can also be seen that Goursat's relation ${\displaystyle k=e^{\varphi }}$ corresponds to ${\displaystyle k=e^{\eta }}$ defined in (3c).

He went on to define ${\displaystyle \alpha ,\beta ,\gamma }$ as the direction cosines normal to surface ${\displaystyle S}$ and ${\displaystyle \alpha _{1},\beta _{1},\gamma _{1}}$ as the ones normal to surface ${\displaystyle S_{1}}$, connected by the transformation:[M 27]

{\displaystyle {\begin{aligned}\alpha _{1}&=\pm {\frac {\alpha }{\cosh \varphi -\gamma \sinh \varphi }}&&&\alpha &=\pm {\frac {\alpha _{1}}{\cosh \varphi +\gamma _{1}\sinh \varphi }}\\\beta _{1}&=\pm {\frac {\beta }{\cosh \varphi -\gamma \sinh \varphi }}&&&\beta &=\pm {\frac {\beta _{1}}{\cosh \varphi +\gamma _{1}\sinh \varphi }}\\\gamma _{1}&=\pm {\frac {\gamma \cosh \varphi -\sinh \varphi }{\cosh \varphi -\gamma \sinh \varphi }}&&&\gamma &=\pm {\frac {\gamma _{1}\cosh \varphi +\sinh \varphi }{\cosh \varphi +\gamma _{1}\sinh \varphi }}\end{aligned}}}
This is equivalent to Lorentz transformation (3e-A) with ${\displaystyle \left[\alpha ,\beta ,\gamma \right]=\left[u_{2},u_{3},u_{1}\right]}$.

### Lindemann (1890–91) – Weierstrass coordinates and Cayley absolute

w:Ferdinand von Lindemann discussed hyperbolic geometry in terms of the w:Cayley–Klein metric in his (1890/91) edition of the lectures on geometry of w:Alfred Clebsch. Citing E:Killing (1885) and Poincaré (1887) in relation to the hyperboloid model in terms of Weierstrass coordinates for the hyperbolic plane and space, he set[M 28]

${\displaystyle {\begin{matrix}\Omega _{xx}=x_{1}^{2}+x_{2}^{2}-4k^{2}x_{3}^{2}=-4k^{2}\ {\text{and}}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}-4k^{2}dx_{3}^{2}\\\Omega _{xx}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=-4k^{2}\ {\text{and}}\ ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-4k^{2}dx_{4}^{2}\end{matrix}}}$

and used the following transformation[M 29]

{\displaystyle {\begin{matrix}X_{1}X_{4}+X_{2}X_{3}=0\\X_{1}X_{4}+X_{2}X_{3}=\Xi _{1}\Xi _{4}+\Xi _{2}\Xi _{3}\\\hline {\begin{aligned}X_{1}&=\left(\lambda +\lambda _{1}\right)U_{4}&\Xi _{1}&=\left(\lambda -\lambda _{1}\right)U_{4}&X_{1}&={\frac {\lambda +\lambda _{1}}{\lambda -\lambda _{1}}}\Xi _{1}\\X_{2}&=\left(\lambda +\lambda _{3}\right)U_{4}&\Xi _{2}&=\left(\lambda -\lambda _{3}\right)U_{4}&X_{2}&={\frac {\lambda +\lambda _{3}}{\lambda -\lambda _{3}}}\Xi _{2}\\X_{3}&=\left(\lambda -\lambda _{3}\right)U_{2}&\Xi _{3}&=\left(\lambda +\lambda _{3}\right)U_{2}&X_{3}&={\frac {\lambda -\lambda _{3}}{\lambda +\lambda _{3}}}\Xi _{3}\\X_{4}&=\left(\lambda -\lambda _{1}\right)U_{1}&\Xi _{4}&=\left(\lambda +\lambda _{1}\right)U_{1}&X_{4}&={\frac {\lambda -\lambda _{1}}{\lambda +\lambda _{1}}}\Xi _{4}\end{aligned}}\end{matrix}}}

into which he put[M 30]

{\displaystyle {\begin{aligned}X_{1}&=x_{1}+2kx_{4},&X_{2}&=x_{2}+ix_{3},&\lambda +\lambda _{1}&=\left(\lambda -\lambda _{1}\right)e^{a},\\X_{4}&=x_{1}-2kx_{4},&X_{3}&=x_{2}-ix_{3},&\lambda +\lambda _{3}&=\left(\lambda -\lambda _{3}\right)e^{\alpha i},\end{aligned}}}
This is equivalent to Lorentz boost (3c) with ${\displaystyle e^{\alpha i}=1}$ and 2k=1 .

From that, he obtained the following Cayley absolute and the corresponding most general motion in hyperbolic space comprising ordinary rotations (a=0) or translations (α=0):[M 30]

{\displaystyle {\begin{matrix}x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-4k^{2}x_{4}^{2}=0\\\hline {\begin{aligned}x_{2}&=\xi _{2}\cos \alpha +\xi _{3}\sin \alpha ,&x_{1}&=\xi _{1}\cos {\frac {a}{i}}+2ki\xi _{4}\sin {\frac {a}{i}},\\x_{3}&=-\xi _{2}\sin \alpha +\xi _{3}\cos \alpha ,&2kx_{4}&=i\xi _{1}\sin {\frac {a}{i}}+2k\xi _{4}\cos {\frac {a}{i}}.\end{aligned}}\end{matrix}}}
This is equivalent to Lorentz boost (3b) with α=0 and 2k=1.

### Gérard (1892) – Weierstrass coordinates

w:Louis Gérard (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane and gave the case of translation as follows:[M 31]

{\displaystyle {\begin{aligned}X&=Z_{0}X'+X_{0}Z'\\Y&=Y'\\Z&=X_{0}X'+Z_{0}Z'\end{aligned}}\ {\text{with}}\ {\begin{aligned}X_{0}&=\operatorname {sh} OO'\\Z_{0}&=\operatorname {ch} OO'\end{aligned}}}
This is equivalent to Lorentz boost (3b).

### Killing (1893,97) – Weierstrass coordinates

w:Wilhelm Killing (1878–1880) gave case of translation in the form[M 32]

${\displaystyle y_{0}=x_{0}\operatorname {Ch} a+x_{1}\operatorname {Sh} a,\quad y_{1}=x_{0}\operatorname {Sh} a+x_{1}\operatorname {Ch} a,\quad y_{2}=x_{2}}$
This is equivalent to Lorentz boost (3b).

In 1898, Killing wrote that relation in a form similar to Escherich (1874), and derived the corresponding Lorentz transformation for the two cases were v is unchanged or u is unchanged:[M 33]

${\displaystyle {\begin{matrix}\xi '={\frac {\xi \operatorname {Ch} {\frac {\mu }{l}}+l\operatorname {Sh} {\frac {\mu }{l}}}{{\frac {\xi }{l}}\operatorname {Sh} {\frac {\mu }{l}}+\operatorname {Ch} {\frac {\mu }{l}}}},\ \eta '={\frac {\eta }{{\frac {\xi }{l}}\operatorname {Sh} {\frac {\mu }{l}}+\operatorname {Ch} {\frac {\mu }{l}}}}\\\hline {\frac {u}{p}}=\xi ,\ {\frac {v}{p}}=\eta \\\hline p'=p\operatorname {Ch} {\frac {\mu }{l}}+{\frac {u}{l}}\operatorname {Sh} {\frac {\mu }{l}},\quad u'=pl\operatorname {Sh} {\frac {\mu }{l}}+u\operatorname {Ch} {\frac {\mu }{l}},\quad v'=v\\{\text{or}}\\p'=p\operatorname {Ch} {\frac {\nu }{l}}+{\frac {v}{l}}\operatorname {Sh} {\frac {\nu }{l}},\quad u'=u,\quad v'=pl\operatorname {Sh} {\frac {\nu }{l}}+v\operatorname {Ch} {\frac {\nu }{l}}\end{matrix}}}$
The upper transformation system is equivalent to Lorentz transformation (3e) and the velocity addition E:(4d) with l=c and ${\displaystyle \mu =c\operatorname {atanh} {\tfrac {v}{c}}}$, the system below is equivalent to Lorentz boost (3b).

### Whitehead (1897/98) – Universal algebra

w:Alfred North Whitehead (1898) discussed the kinematics of hyperbolic space as part of his study of w:universal algebra, and obtained the following transformation:[M 34]

{\displaystyle {\begin{aligned}x'&=\left(\eta \cosh {\frac {\delta }{\gamma }}+\eta _{1}\sinh {\frac {\delta }{\gamma }}\right)e+\left(\eta \sinh {\frac {\delta }{\gamma }}+\eta _{1}\cosh {\frac {\delta }{\gamma }}\right)e_{1}\\&\qquad +\left(\eta _{2}\cos \alpha +\eta _{3}\sin \alpha \right)e_{2}+\left(\eta _{3}\cos \alpha -\eta _{2}\sin \alpha \right)e_{3}\end{aligned}}}
This is equivalent to Lorentz boost (3b) with α=0.

### Elliott (1903) – Invariant theory

w:Edwin Bailey Elliott (1903) discussed a special cyclical subgroup of ternary linear transformations for which the (unit) determinant of transformation is resoluble into three ordinary algebraical factors, which he pointed out is in direct analogy to a subgroup formed by the following transformations:[M 35]

${\displaystyle {\begin{matrix}x=X\cosh \phi +Y\sinh \phi ,\quad y=X\sinh \phi +Y\cosh \phi \\\hline X+Y=e^{-\phi }(x+y),\quad X-Y=e^{\phi }(x-y)\end{matrix}}}$
This is equivalent to Lorentz boost (3b) and (3c). The mentioned subgroup corresponds to the one-parameter subgroup generated by Lorentz boosts.

### Woods (1903) – Weierstrass coordinates

w:Frederick S. Woods (1903, published 1905) gave the case of translation in hyperbolic space:[M 36]

${\displaystyle x_{1}^{\prime }=x_{1}\cos kl+x_{0}{\frac {\sin kl}{k}},\quad x_{2}^{\prime }=x_{2},\quad x_{2}^{\prime }=x_{3},\quad x_{0}^{\prime }=-x_{1}k\sin kl+x_{0}\cos kl}$
This is equivalent to Lorentz boost (3b) with k2=-1.

and the loxodromic substitution for hyperbolic space:[M 37]

{\displaystyle {\begin{matrix}{\begin{aligned}x_{1}^{\prime }&=x_{1}\cosh \alpha -x_{0}\sinh \alpha \\x_{2}^{\prime }&=x_{2}\cos \beta -x_{3}\sin \beta \\x_{3}^{\prime }&=x_{2}\sin \beta +x_{3}\cos \beta \\x_{0}^{\prime }&=-x_{1}\sinh \alpha +x_{0}\cosh \alpha \end{aligned}}\end{matrix}}}
This is equivalent to Lorentz boost (3b) with β=0.

### Liebmann (1904–05) – Weierstrass coordinates

w:Heinrich Liebmann (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – gave the case of translation in the hyperbolic plane:[M 38]

${\displaystyle x_{1}^{\prime }=x'\operatorname {ch} a+p'\operatorname {sh} a,\quad y_{1}^{\prime }=y',\quad p_{1}^{\prime }=x'\operatorname {sh} a+p'\operatorname {ch} a}$
This is equivalent to Lorentz boost (3b).

### Frank (1909) – Special relativity

In special relativity, hyperbolic functions were used by w:Philipp Frank (1909), who derived the Lorentz transformation using ψ as rapidity:[R 2]

${\displaystyle {\begin{matrix}x'=x\varphi (a)\,{\rm {ch}}\,\psi +t\varphi (a)\,{\rm {sh}}\,\psi \\t'=-x\varphi (a)\,{\rm {sh}}\,\psi +t\varphi (a)\,{\rm {ch}}\,\psi \\\hline {\rm {th}}\,\psi =-a,\ {\rm {sh}}\,\psi ={\frac {a}{\sqrt {1-a^{2}}}},\ {\rm {ch}}\,\psi ={\frac {1}{\sqrt {1-a^{2}}}},\ \varphi (a)=1\\\hline x'={\frac {x-at}{\sqrt {1-a^{2}}}},\ y'=y,\ z'=z,\ t'={\frac {-ax+t}{\sqrt {1-a^{2}}}}\end{matrix}}}$
This is equivalent to Lorentz boost (3b).

### Herglotz (1909/10) – Special relativity

In special relativity, w:Gustav Herglotz (1909/10) classified the one-parameter Lorentz transformations as loxodromic, hyperbolic, parabolic and elliptic, with the hyperbolic case being:[R 3]

{\displaystyle {\begin{matrix}Z=Z'e^{\vartheta }\\{\begin{aligned}x&=x',&t-z&=(t'-z')e^{\vartheta }\\y&=y',&t+z&=(t'+z')e^{-\vartheta }\end{aligned}}\end{matrix}}}
This is equivalent to Lorentz boost (3c).

### Varićak (1910) – Special relativity

In special relativity, hyperbolic functions were used by w:Vladimir Varićak in several papers starting from 1910, who represented the equations of special relativity on the basis of w:hyperbolic geometry in terms of Weierstrass coordinates. For instance, by setting l=ct and v/c=tanh(u) with u as rapidity he wrote the Lorentz transformation in agreement with (4b):[R 4]

{\displaystyle {\begin{aligned}l'&=-x\operatorname {sh} u+l\operatorname {ch} u,\\x'&=x\operatorname {ch} u-l\operatorname {sh} u,\\y'&=y,\quad z'=z,\\\operatorname {ch} u&={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}\end{aligned}}}
This is equivalent to Lorentz boost (3b).

He showed the relation of rapidity to the w:Gudermannian function and the w:angle of parallelism:[R 4]

${\displaystyle {\frac {v}{c}}=\operatorname {th} u=\operatorname {tg} \psi =\sin \operatorname {gd} (u)=\cos \Pi (u)}$

He also related the velocity addition to the w:hyperbolic law of cosines:[R 5]

${\displaystyle {\begin{matrix}\operatorname {ch} {u}=\operatorname {ch} {u_{1}}\operatorname {c} h{u_{2}}+\operatorname {sh} {u_{1}}\operatorname {sh} {u_{2}}\cos \alpha \\\operatorname {ch} {u_{i}}={\frac {1}{\sqrt {1-\left({\frac {v_{i}}{c}}\right)^{2}}}},\ \operatorname {sh} {u_{i}}={\frac {v_{i}}{\sqrt {1-\left({\frac {v_{i}}{c}}\right)^{2}}}}\\v={\sqrt {v_{1}^{2}+v_{2}^{2}-\left({\frac {v_{1}v_{2}}{c}}\right)^{2}}}\ \left(a={\frac {\pi }{2}}\right)\end{matrix}}}$
This is equivalent to Lorentz boost (3f).

## References

### Historical mathematical sources

1. Mercator (1667), prop. XIV, pp. 28-29. (He used this result to derive the Mercator series in prop. XV).
2. Kepler (1609), chapter 60. The editors of Kepler's collected papers remark (p. 482), that Kepler's relations correspond to ${\displaystyle {\scriptstyle \alpha =\beta +e\sin \beta }}$ and ${\displaystyle {\scriptstyle \cos \nu ={\frac {e+\cos \beta }{1+e\cos \beta }}}}$ and ${\displaystyle {\scriptstyle \cos \beta ={\frac {\cos \nu -e}{1-e\cos \nu }}}}$
3. Euler (1735/40), § 19
4. Euler (1748a), section VIII
5. Lagrange (1770/71), section I
6. Riccati (1757), p. 71
7. Günther (1880/81), pp. 7–13
8. Lambert (1761/68), pp. 309–318
9. Lambert (1770), p. 335
10. Taurinus (1826), p. 66; see also p. 272 in the translation by Engel and Stäckel (1899)
11. Beltrami (1868a), pp. 287-288; Note I; Note II
12. Beltrami (1868b), pp. 232, 240–241, 253–254
13. Laisant (1874b), pp. 134–135
14. Gudermann (1830), §1–3, §18–19
15. Escherich (1874), p. 508
16. Escherich (1874), p. 510
17. Cox (1881), p. 186
18. Glaisher (1878), p. 30
19. Günther (1880/81), p. 405
20. Cox (1881/82), p. 194
21. Lipschitz (1886), pp. 90–92
22. Schur (1885/86), p. 167
23. Schur (1900/02), p. 290; (1909), p. 83
24. Schur (1900/02), p. 291; (1909), p. 83
25. Goursat (1887/88), p. 144
26. Goursat (1887/88), p. 145
27. Goursat (1887/88), p. 149f.
28. Lindemann & Clebsch (1890/91), pp. 477–478, 524
29. Lindemann & Clebsch (1890/91), pp. 361–362
30. Lindemann & Clebsch (1890/91), p. 496
31. Gérard (1892), pp. 40–41
32. Killing (1893), p. 331
33. Killing (1898), p. 133
35. Elliott (1903), p. 109
36. Woods (1903/05), p. 55
37. Woods (1903/05), p. 72
38. Liebmann (1904/05), p. 174

### Historical relativity sources

1. Varićak (1912), p. 108
2. Frank (1909), pp. 423-425
3. Herglotz (1909/10), pp. 404-408
4. Varićak (1910), p. 93
5. Varićak (1910), p. 94
• Varićak, V. (1910), "Anwendung der Lobatschefskijschen Geometrie in der Relativtheorie", Physikalische Zeitschrift, 11: 93–6
• Varičak, V. (1912), "Über die nichteuklidische Interpretation der Relativtheorie", Jahresbericht der Deutschen Mathematiker-Vereinigung, 21: 103–127

### Secondary sources

1. Rindler (1969), p. 45
2. Rosenfeld (1988), p. 231
3. Pauli (1921), p. 561
4. Barrett (2006), chapter 4, section 2
5. Volk (1976), p. 366
6. Barnett (2004), pp. 22–23
7. Bonola (1912), p. 79
8. Gray (1979), p. 242
9. Sommerville (1911), p. 297
English translation: Pauli, W. (1981) [1958], "Theory of Relativity", Fundamental Theories of Physics, Dover Publications, 165, ISBN 978-0-486-64152-2
• Rindler, W. (2013) [1969], Essential Relativity: Special, General, and Cosmological, Springer, ISBN 978-1475711356
• Rosenfeld, B.A. (1988), A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space, New York: Springer, ISBN 978-1441986801
• Sommerville, D. M. L. Y. (1911), Bibliography of non-Euclidean geometry, London: London Pub. by Harrison for the University of St. Andrews
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