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

## Most general Lorentz transformations

The general w:quadratic form q(x) with coefficients of a w:symmetric matrix A, the associated w:bilinear form b(x,y), and the w:linear transformations of q(x) and b(x,y) into q(x′) and b(x′,y′) using the w:transformation matrix g, can be written as

{\begin{matrix}{\begin{aligned}{\begin{aligned}q=\sum _{0}^{n}A_{ij}x_{i}x_{j}=\mathbf {x} ^{\mathrm {T} }\cdot \mathbf {A} \cdot \mathbf {x} \end{aligned}}&=q'=\mathbf {x} ^{\mathrm {\prime T} }\cdot \mathbf {A} '\cdot \mathbf {x} '\\b=\sum _{0}^{n}A_{ij}x_{i}y_{j}=\mathbf {x} ^{\mathrm {T} }\cdot \mathbf {A} \cdot \mathbf {y} &=b'=\mathbf {x} ^{\mathrm {\prime T} }\cdot \mathbf {A} '\cdot \mathbf {y} '\end{aligned}}\quad \left(A_{ij}=A_{ji}\right)\\\hline \left.{\begin{aligned}x_{i}^{\prime }&=\sum _{j=0}^{n}g_{ij}x_{j}=\mathbf {g} \cdot \mathbf {x} \\x_{i}&=\sum _{j=0}^{n}g_{ij}^{(-1)}x_{j}^{\prime }=\mathbf {g} ^{-1}\cdot \mathbf {x} '\end{aligned}}\right|\mathbf {g} ^{\rm {T}}\cdot \mathbf {A} \cdot \mathbf {g} =\mathbf {A} '\end{matrix}} (Q1)

The case n=1 is the w:binary quadratic form introduced by Lagrange (1773) and Gauss (1798/1801), n=2 is the ternary quadratic form introduced by Gauss (1798/1801), n=3 is the quaternary quadratic form etc.

### Most general Lorentz transformation

The general Lorentz transformation follows from (Q1) by setting A=A′=diag(-1,1,...,1) and det g=±1. It forms an w:indefinite orthogonal group called the w:Lorentz group O(1,n), while the case det g=+1 forms the restricted w:Lorentz group SO(1,n). The quadratic form q(x) becomes the w:Lorentz interval in terms of an w:indefinite quadratic form of w:Minkowski space (being a special case of w:pseudo-Euclidean space), and the associated bilinear form b(x) becomes the w:Minkowski inner product:

{\begin{matrix}{\begin{aligned}-x_{0}^{2}+\cdots +x_{n}^{2}&=-x_{0}^{\prime 2}+\dots +x_{n}^{\prime 2}\\-x_{0}y_{0}+\cdots +x_{n}y_{n}&=-x_{0}^{\prime }y_{0}^{\prime }+\cdots +x_{n}^{\prime }y_{n}^{\prime }\end{aligned}}\\\hline \left.{\begin{matrix}\mathbf {x} '=\mathbf {g} \cdot \mathbf {x} \\\downarrow \\{\begin{aligned}x_{0}^{\prime }&=x_{0}g_{00}+x_{1}g_{01}+\dots +x_{n}g_{0n}\\x_{1}^{\prime }&=x_{0}g_{10}+x_{1}g_{11}+\dots +x_{n}g_{1n}\\&\dots \\x_{n}^{\prime }&=x_{0}g_{n0}+x_{1}g_{n1}+\dots +x_{n}g_{nn}\end{aligned}}\\\\\mathbf {x} =\mathbf {g} ^{-1}\cdot \mathbf {x} '\\\downarrow \\{\begin{aligned}x_{0}&=x_{0}^{\prime }g_{00}-x_{1}^{\prime }g_{10}-\dots -x_{n}^{\prime }g_{n0}\\x_{1}&=-x_{0}^{\prime }g_{01}+x_{1}^{\prime }g_{11}+\dots +x_{n}^{\prime }g_{n1}\\&\dots \\x_{n}&=-x_{0}^{\prime }g_{0n}+x_{1}^{\prime }g_{1n}+\dots +x_{n}^{\prime }g_{nn}\end{aligned}}\end{matrix}}\right|{\begin{matrix}{\begin{aligned}\mathbf {A} \cdot \mathbf {g} ^{\mathrm {T} }\cdot \mathbf {A} &=\mathbf {g} ^{-1}\\\mathbf {g} ^{\rm {T}}\cdot \mathbf {A} \cdot \mathbf {g} &=\mathbf {A} \\\mathbf {g} \cdot \mathbf {A} \cdot \mathbf {g} ^{\mathrm {T} }&=\mathbf {A} \\\\\end{aligned}}\\{\begin{aligned}\sum _{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k}&=\left\{{\begin{aligned}-1\quad &(j=k=0)\\1\quad &(j=k>0)\\0\quad &(j\neq k)\end{aligned}}\right.\\\sum _{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0}&=\left\{{\begin{aligned}-1\quad &(i=k=0)\\1\quad &(i=k>0)\\0\quad &(i\neq k)\end{aligned}}\right.\end{aligned}}\end{matrix}}\end{matrix}} (1a)

Such general Lorentz transformations (1a) for various dimensions were used by Gauss (1818), Jacobi (1827, 1833), Lebesgue (1837), Bour (1856), Somov (1863), Hill (1882) in order to simplify computations of w:elliptic functions and integrals. They were also used by Poincaré (1881), Cox (1881/82), Picard (1882, 1884), Killing (1885, 1893), Gérard (1892), Hausdorff (1899), Woods (1901, 1903), Liebmann (1904/05) to describe w:hyperbolic motions (i.e. rigid motions in the w:hyperbolic plane or w:hyperbolic space), which were expressed in terms of Weierstrass coordinates of the w:hyperboloid model satisfying the relation $-x_{0}^{2}+\cdots +x_{n}^{2}=-1$ or in terms of the w:Cayley–Klein metric of w:projective geometry using the "absolute" form $-x_{0}^{2}+\cdots +x_{n}^{2}=0$ .[M 1] In addition, w:infinitesimal transformations related to the w:Lie algebra of the group of hyperbolic motions were given in terms of Weierstrass coordinates $-x_{0}^{2}+\cdots +x_{n}^{2}=-1$ by Killing (1888-1897).

### Most general Lorentz transformation of velocity

If $x_{i},\ x_{i}^{\prime }$ in (1a) are interpreted as w:homogeneous coordinates, then the corresponding inhomogenous coordinates $u_{s},\ u_{s}^{\prime }$ follow by

$\left[{\frac {x_{0}}{x_{0}}},\ {\frac {x_{s}}{x_{0}}}\right]=\left[1,\ u_{s}\right],\ \left[{\frac {x_{0}^{\prime }}{x_{0}^{\prime }}},\ {\frac {x_{s}^{\prime }}{x_{0}^{\prime }}}\right]=\left[1,\ u_{s}^{\prime }\right],\ (s=1,2\dots n)$ so that the Lorentz transformation becomes a w:homography leaving invariant the equation of the w:unit sphere, which w:John Lighton Synge called "the most general formula for the composition of velocities" in terms of special relativity (the transformation matrix g stays the same as in (1a)):

{\begin{matrix}{\begin{matrix}-x_{0}^{2}+\cdots +x_{n}^{2}=-x_{0}^{\prime 2}+\dots +x_{n}^{\prime 2}&\rightarrow &{\begin{aligned}-1+u_{1}^{2}+\cdots +u_{n}^{2}&={{\frac {-1+u_{1}^{\prime 2}+\cdots +u_{n}^{\prime 2}}{\left(g_{00}+g_{01}u_{1}^{\prime }+\dots +g_{0n}u_{n}^{\prime }\right)^{2}}}}\\{{\frac {-1+u_{1}^{2}+\cdots +u_{n}^{2}}{\left(g_{00}-g_{10}u_{1}-\dots -g_{n0}u_{n}\right)^{2}}}}&=-1+u_{1}^{\prime 2}+\cdots +u_{n}^{\prime 2}\end{aligned}}\\\hline -x_{0}^{2}+\cdots +x_{n}^{2}=-x_{0}^{\prime 2}+\dots +x_{n}^{\prime 2}=0&\rightarrow &-1+u_{1}^{2}+\cdots +u_{n}^{2}=-1+u_{1}^{\prime 2}+\cdots +u_{n}^{\prime 2}=0\end{matrix}}\\\hline {\begin{aligned}u_{s}^{\prime }&={\frac {g_{s0}+g_{s1}u_{1}+\dots +g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots +g_{0n}u_{n}}}\\\\u_{s}&={\frac {-g_{0s}+g_{1s}u_{1}^{\prime }+\dots +g_{ns}u_{n}^{\prime }}{g_{00}-g_{10}u_{1}^{\prime }-\dots -g_{n0}u_{n}^{\prime }}}\end{aligned}}\left|{\begin{aligned}\sum _{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k}&=\left\{{\begin{aligned}-1\quad &(j=k=0)\\1\quad &(j=k>0)\\0\quad &(j\neq k)\end{aligned}}\right.\\\sum _{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0}&=\left\{{\begin{aligned}-1\quad &(i=k=0)\\1\quad &(i=k>0)\\0\quad &(i\neq k)\end{aligned}}\right.\end{aligned}}\right.\end{matrix}} (1b)

Such Lorentz transformations for various dimensions were used by Gauss (1818), Jacobi (1827–1833), Lebesgue (1837), Bour (1856), Somov (1863), Hill (1882), Callandreau (1885) in order to simplify computations of elliptic functions and integrals, by Picard (1882-1884) in relation to Hermitian quadratic forms, or by Woods (1901, 1903) in terms of the w:Beltrami–Klein model of hyperbolic geometry. In addition, infinitesimal transformations in terms of the w:Lie algebra of the group of hyperbolic motions leaving invariant the unit sphere $-1+u_{1}^{\prime 2}+\cdots +u_{n}^{\prime 2}=0$ were given by Lie (1885-1893) and Werner (1889) and Killing (1888-1897).

## Historical notation

### Lagrange (1773) – Binary quadratic forms

After the invariance of the sum of squares under linear substitutions was discussed by Euler (1771), the general expressions of a w:binary quadratic form and its transformation was formulated by w:Joseph-Louis Lagrange (1773/75) as follows[M 2]

{\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\\hline {\begin{aligned}y&=Ms+Nx\\z&=ms+nx\end{aligned}}\left|{\begin{matrix}{\begin{aligned}P&=pM^{2}+2qMm+rm^{2}\\Q&=pMN+q(Mn+Nm)+rmn\\R&=pN^{2}+2qNn+rn^{2}\end{aligned}}\\\downarrow \\PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2}\end{matrix}}\right.\end{matrix}} which is equivalent to (Q1) (n=1).

The Lorentz interval $-x_{0}^{2}+x_{1}^{2}$ and the Lorentz transformation (1a) (n=1) are a special case of the binary quadratic form by setting (p,q,r)=(P,Q,R)=(1,0,-1).

### Gauss (1798–1818)

The theory of binary quadratic forms was considerably expanded by w:Carl Friedrich Gauss (1798, published 1801) in his w:Disquisitiones Arithmeticae. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:[M 3]

{\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\F'=a'x^{\prime 2}+2b'x'y'+c'y^{\prime 2}=(a',b',c')\\\hline {\begin{aligned}x&=\alpha x'+\beta y'\\y&=\gamma x'+\delta y'\\\\x'&=\delta x-\beta y\\y'&=-\gamma x+\alpha y\end{aligned}}\left|{\begin{matrix}{\begin{aligned}a'&=a\alpha ^{2}+2b\alpha \gamma +c\gamma ^{2}\\b'&=a\alpha \beta +b(\alpha \delta +\beta \gamma )+c\gamma \delta \\c'&=a\beta ^{2}+2b\beta \delta +c\delta ^{2}\end{aligned}}\\\downarrow \\b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha \delta -\beta \gamma )^{2}\end{matrix}}\right.\end{matrix}} which is equivalent to (Q1) (n=1). As pointed out by Gauss, F and F′ are called "proper equivalent" if αδ-βγ=1, so that F is contained in F′ as well as F′ is contained in F. In addition, if another form F″ is contained by the same procedure in F′ it is also contained in F and so forth.[M 4]

The Lorentz interval $-x_{0}^{2}+x_{1}^{2}$ and the Lorentz transformation (1a) (n=1) are a special case of the binary quadratic form by setting (a,b,c)=(a',b',c')=(1,0,-1).

Gauss (1798/1801)[M 5] also discussed ternary quadratic forms with the general expression

{\begin{matrix}f=ax^{2}+a'x^{\prime 2}+a''x^{\prime \prime 2}+2bx'x''+2b'xx''+2b''xx'=\left({\begin{matrix}a,&a',&a''\\b,&b',&b''\end{matrix}}\right)\\g=my^{2}+m'y^{\prime 2}+m''y^{\prime \prime 2}+2ny'y''+2n'yy''+2n''yy'=\left({\begin{matrix}m,&m',&m''\\n,&n',&n''\end{matrix}}\right)\\\hline {\begin{aligned}x&=\alpha y+\beta y'+\gamma y''\\x'&=\alpha 'y+\beta 'y'+\gamma 'y''\\x''&=\alpha ''y+\beta ''y'+\gamma ''y''\end{aligned}}\end{matrix}} which is equivalent to (Q1) (n=2). Gauss called these forms definite when they have the same sign such as x2+y2+z2, or indefinite in the case of different signs such as x2+y2-z2. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:[M 6]

$\left({\begin{matrix}a,&a',&a''\\b,&b',&b''\end{matrix}}\right)\Rightarrow {\begin{matrix}\left({\begin{matrix}1,&-1,&1\\0,&0,&0\end{matrix}}\right),\ \left({\begin{matrix}-1,&1,&1\\0,&0,&0\end{matrix}}\right),\ \left({\begin{matrix}1,&1,&-1\\0,&0,&0\end{matrix}}\right),\\\left({\begin{matrix}1,&-1,&-1\\0,&0,&0\end{matrix}}\right),\ \left({\begin{matrix}-1,&1,&-1\\0,&0,&0\end{matrix}}\right),\ \left({\begin{matrix}-1,&-1,&1\\0,&0,&0\end{matrix}}\right)\end{matrix}}$ These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval $x^{2}+x^{\prime 2}-x^{\prime \prime 2}$ and the Lorentz transformation (1a) (n=2) is an indefinite ternary quadratic form, which follows from the general ternary form by setting:
$\left({\begin{matrix}a,&a',&a''\\b,&b',&b''\end{matrix}}\right)=\left({\begin{matrix}m,&m',&m''\\n,&n',&n''\end{matrix}}\right)=\left({\begin{matrix}1,&1,&-1\\0,&0,&0\end{matrix}}\right)$ #### Homogeneous coordinates

Gauss (1818) discussed planetary motions together with formulating w:elliptic functions. In order to simplify the integration, he transformed the expression

$(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E$ into

$G+G'\cos T^{2}+G''\sin T^{2}$ in which the w:eccentric anomaly E is connected to the new variable T by the following transformation including an arbitrary constant k, which Gauss then rewrote by setting k=1:[M 7]

{\begin{matrix}{\left(\alpha +\alpha '\cos T+\alpha ''\sin T\right)^{2}+\left(\beta +\beta '\cos T+\beta ''\sin T\right)^{2}-\left(\gamma +\gamma '\cos T+\gamma ''\sin T\right)^{2}}=0\\k\left(\cos ^{2}T+\sin ^{2}T-1\right)=0\\\hline {\begin{aligned}\cos E&={\frac {\alpha +\alpha '\cos T+\alpha ''\sin T}{\gamma +\gamma '\cos T+\gamma ''\sin T}}\\\sin E&={\frac {\beta +\beta '\cos T+\beta ''\sin T}{\gamma +\gamma '\cos T+\gamma ''\sin T}}\end{aligned}}\left|{{\begin{aligned}-\alpha \alpha -\beta \beta +\gamma \gamma &=k&\alpha \alpha -\alpha '\alpha '-\alpha ''\alpha ''&=-k\\-\alpha '\alpha '-\beta '\beta '+\gamma '\gamma '&=-k&\beta \beta -\beta '\beta '-\beta ''\beta ''&=-k\\-\alpha ''\alpha ''-\beta ''\beta ''+\gamma ''\gamma ''&=-k&\gamma \gamma -\gamma '\gamma '-\gamma ''\gamma ''&=+k\\-\alpha '\alpha ''-\beta '\beta ''+\gamma '\gamma ''&=0&\beta \gamma -\beta '\gamma '-\beta ''\gamma ''&=0\\-\alpha ''\alpha -\beta ''\beta +\gamma ''\gamma &=0&\gamma \alpha -\gamma '\alpha '-\gamma ''\alpha ''&=0\\-\alpha \alpha '-\beta \beta '+\gamma \gamma '&=0&\alpha \beta -\alpha '\beta '-\alpha ''\beta ''&=0\end{aligned}}}\right.\\\hline k=1\\{\begin{aligned}t\cos E&=\alpha +\alpha '\cos T+\alpha ''\sin T\\t\sin E&=\beta +\beta '\cos T+\beta ''\sin T\\t&=\gamma +\gamma '\cos T+\gamma ''\sin T\end{aligned}}\left|{{\begin{aligned}-\alpha \alpha -\beta \beta +\gamma \gamma &=1\\-\alpha '\alpha '-\beta '\beta '+\gamma '\gamma '&=-1\\-\alpha ''\alpha ''-\beta ''\beta ''+\gamma ''\gamma ''&=-1\\-\alpha '\alpha ''-\beta '\beta ''+\gamma '\gamma ''&=0\\-\alpha ''\alpha -\beta ''\beta +\gamma ''\gamma &=0\\-\alpha \alpha '-\beta \beta '+\gamma \gamma '&=0\end{aligned}}}\right.\end{matrix}} The coefficients α,β,γ,... of Gauss' case k=1 are equivalent to the coefficient system in Lorentz transformations (1a) and (1b) (n=2). Further setting $[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime },\ u_{2}^{\prime }\right]$ , Gauss' transformation becomes Lorentz transformation (1b) (n=2).

Subsequently, he showed that these relations can be reformulated using three variables x,y,z and u,u′,u″, so that

$aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz$ can be transformed into

$Guu+G'u'u'+G''u''u''$ ,

in which x,y,z and u,u′,u″ are related by the transformation:[M 8]

{\begin{aligned}x&=\alpha u+\alpha 'u'+\alpha ''u''\\y&=\beta u+\beta 'u'+\beta ''u''\\z&=\gamma u+\gamma 'u'+\gamma ''u''\\\\u&=-\alpha x-\beta y+\gamma z\\u'&=\alpha 'x+\beta 'y-\gamma 'z\\u''&=\alpha ''x+\beta ''y-\gamma ''z\end{aligned}}\left|{{\begin{aligned}-\alpha \alpha -\beta \beta +\gamma \gamma &=1\\-\alpha '\alpha '-\beta '\beta '+\gamma '\gamma '&=-1\\-\alpha ''\alpha ''-\beta ''\beta ''+\gamma ''\gamma ''&=-1\\-\alpha '\alpha ''-\beta '\beta ''+\gamma '\gamma ''&=0\\-\alpha ''\alpha -\beta ''\beta +\gamma ''\gamma &=0\\-\alpha \alpha '-\beta \beta '+\gamma \gamma '&=0\end{aligned}}}\right. This is equivalent to Lorentz transformation (1a) (n=2) satisfying $x^{2}+y^{2}-z^{2}=u^{\prime 2}+u^{\prime \prime 2}-u^{2}$ , and can be related to Gauss' previous equations in terms of homogeneous coordinates $\left[\cos T,\sin T,\cos E,\sin E\right]=\left[{\tfrac {x}{z}},\ {\tfrac {y}{z}},\ {\tfrac {u'}{u}},\ {\tfrac {u''}{u}}\right]$ .

### Jacobi (1827, 1833/34) – Homogeneous coordinates

Following Gauss (1818), w:Carl Gustav Jacob Jacobi extended Gauss' transformation to 3 dimensions in 1827:[M 9]

{{\begin{matrix}\cos P^{2}+\sin P^{2}\cos \vartheta ^{2}+\sin P^{2}\sin \vartheta ^{2}=1\\k\left(\cos \psi ^{2}+\sin \psi ^{2}\cos \varphi ^{2}+\sin \psi ^{2}\sin \varphi ^{2}-1\right)=0\\\hline {\left.{\begin{matrix}\mathbf {(1)} {\begin{aligned}\cos P&={\frac {\alpha +\alpha '\cos \psi +\alpha ''\sin \psi \cos \varphi +\alpha '''\sin \psi \sin \varphi }{\delta +\delta '\cos \psi +\delta ''\sin \psi \cos \varphi +\delta '''\sin \psi \sin \varphi }}\\\sin P\cos \vartheta &={\frac {\beta +\beta '\cos \psi +\beta ''\sin \psi \cos \varphi +\beta '''\sin \psi \sin \varphi }{\delta +\delta '\cos \psi +\delta ''\sin \psi \cos \varphi +\delta '''\sin \psi \sin \varphi }}\\\sin P\sin \vartheta &={\frac {\gamma +\beta '\cos \psi +\gamma ''\sin \psi \cos \varphi +\gamma '''\sin \psi \sin \varphi }{\delta +\delta '\cos \psi +\delta ''\sin \psi \cos \varphi +\delta '''\sin \psi \sin \varphi }}\\\\\cos \psi &={\frac {-\delta '+\alpha '\cos P+\beta '\sin P\cos \vartheta +\gamma '\sin P\sin \vartheta }{\delta -\alpha \cos P-\beta \sin P\cos \vartheta -\gamma \sin P\sin \vartheta }}\\\sin \psi \cos \varphi &={\frac {-\delta ''+\alpha ''\cos P+\beta ''\sin P\cos \vartheta +\gamma ''\sin P\sin \vartheta }{\delta -\alpha \cos P-\beta \sin P\cos \vartheta -\gamma \sin P\sin \vartheta }}\\\sin \psi \sin \varphi &={\frac {-\delta '''+\alpha '''\cos P+\beta '''\sin P\cos \vartheta +\gamma '''\sin P\sin \vartheta }{\delta -\alpha \cos P-\beta \sin P\cos \vartheta -\gamma \sin P\sin \vartheta }}\end{aligned}}\\\\\hline \mathbf {(2)} {\begin{aligned}\alpha \mu +\beta x+\gamma y+\delta z&=m\\\alpha '\mu +\beta 'x+\gamma 'y+\delta 'z&=m'\\\alpha ''\mu +\beta ''x+\gamma ''y+\delta ''z&=m''\\\alpha '''\mu +\beta '''x+\gamma '''y+\delta '''z&=m'''\\\\Am+A'm'+A''m''+A'''m'''&=\mu \\Bm+B'm'+B''m''+B'''m'''&=x\\Cm+C'm'+C''m''+C'''m'''&=y\\Dm+D'm'+D''m''+D'''m'''&=z\\\\\end{aligned}}\\{\begin{aligned}\alpha &=-kA,&\beta &=-kB,&\gamma &=-kC,&\delta &=kD,\\\alpha '&=kA',&\beta '&=kB',&\gamma '&=kC',&\delta '&=-kD',\\\alpha ''&=kA'',&\beta ''&=kB'',&\gamma ''&=kC'',&\delta ''&=-kD'',\\\alpha '''&=kA''',&\beta '''&=kB''',&\gamma '''&=kC''',&\delta '''&=-kD''',\end{aligned}}\end{matrix}}\right|{\begin{matrix}{\begin{aligned}\alpha \alpha +\beta \beta +\gamma \gamma -\delta \delta &=-k\\\alpha '\alpha '+\beta '\beta '+\gamma '\gamma '-\delta '\delta '&=k\\\alpha ''\alpha ''+\beta ''\beta ''+\gamma ''\gamma ''-\delta ''\delta ''&=k\\\alpha '''\alpha '''+\beta '''\beta '''+\gamma '''\gamma '''-\delta '''\delta '''&=k\\\alpha \alpha '+\beta \beta '+\gamma \gamma '-\delta \delta '&=0\\\alpha \alpha ''+\beta \beta ''+\gamma \gamma ''-\delta \delta ''&=0\\\alpha \alpha '''+\beta \beta '''+\gamma \gamma '''-\delta \delta '''&=0\\\alpha ''\alpha '''+\beta ''\beta '''+\gamma ''\gamma '''-\delta ''\delta '''&=0\\\alpha '''\alpha '+\beta '''\beta '+\gamma '''\gamma '-\delta '''\delta '&=0\\\alpha '\alpha ''+\beta '\beta ''+\gamma '\gamma ''-\delta '\delta ''&=0\\\\-\alpha \alpha +\alpha '\alpha '+\alpha ''\alpha ''+\alpha '''\alpha '''&=k\\-\beta \beta +\beta '\beta '+\beta ''\beta ''+\beta '''\beta '''&=k\\-\gamma \gamma +\gamma '\gamma '+\gamma ''\gamma ''+\gamma '''\gamma '''&=k\\-\delta \delta +\delta '\delta '+\delta ''\delta ''+\delta '''\delta '''&=-k\\-\alpha \beta +\alpha '\beta '+\alpha ''\beta ''+\alpha '''\beta '''&=0\\-\alpha \gamma +\alpha '\gamma '+\alpha ''\gamma ''+\alpha '''\gamma '''&=0\\-\alpha \delta +\alpha '\delta '+\alpha ''\delta ''+\alpha '''\delta '''&=0\\-\beta \gamma +\beta '\gamma '+\beta ''\gamma ''+\beta '''\gamma '''&=0\\-\gamma \delta +\gamma '\delta '+\gamma ''\delta ''+\gamma '''\delta '''&=0\\-\delta \beta +\delta '\beta '+\delta ''\beta ''+\delta '''\beta '''&=0\end{aligned}}\end{matrix}}}\end{matrix}}} By setting {{\begin{aligned}\left[\cos P,\ \sin P\cos \varphi ,\ \sin P\sin \varphi \right]&=\left[u_{1},\ u_{2},\ u_{3}\right]\\{}[\cos \psi ,\ \sin \psi \cos \vartheta ,\ \sin \psi \sin \vartheta ]&=\left[u_{1}^{\prime },\ u_{2}^{\prime },\ u_{3}^{\prime }\right]\end{aligned}}} and k=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation (1b) (n=3), and by setting k=1 in transformation system (2) it becomes equivalent to Lorentz transformation (1a) (n=3) producing $m^{2}+m^{\prime 2}+m^{\prime \prime 2}-m^{\prime \prime \prime 2}=\mu ^{2}+x^{2}+y^{2}-z^{2}$ .

Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change) in the case of 2 dimensions:[M 10]

{{\begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\\mathbf {(1)} {\begin{aligned}x&=\alpha s+\alpha 's'+\alpha ''s''\\y&=\beta s+\beta 's'+\beta ''s''\\z&=\gamma s+\gamma 's'+\gamma ''s''\\\\s&=\alpha x+\beta y+\gamma z\\s'&=\alpha 'x+\beta 'y+\gamma 'z\\s''&=\alpha ''x+\beta ''y+\gamma ''z\end{aligned}}\left|{\begin{aligned}\alpha \alpha +\beta \beta +\gamma \gamma &=1&\alpha \alpha +\alpha '\alpha '+\alpha ''\alpha ''&=1\\\alpha '\alpha '+\beta '\beta '+\gamma '\gamma '&=1&\beta \beta +\beta '\beta '+\beta ''\beta ''&=1\\\alpha ''\alpha ''+\beta ''\beta ''+\gamma ''\gamma ''&=1&\gamma \gamma +\gamma '\gamma '+\gamma ''\gamma ''&=1\\\alpha '\alpha ''+\beta '\beta ''+\gamma '\gamma ''&=0&\beta \gamma +\beta '\gamma '+\beta ''\gamma ''&=0\\\alpha ''\alpha +\beta ''\beta +\gamma ''\gamma &=0&\gamma \alpha +\gamma '\alpha '+\gamma ''\alpha ''&=0\\\alpha \alpha '+\beta \beta '+\gamma \gamma '&=0&\alpha \beta +\alpha '\beta '+\alpha ''\beta ''&=0\end{aligned}}\right.\\\hline \left[{\frac {y}{x}},\ {\frac {z}{x}},\ {\frac {s'}{s}},\ {\frac {s''}{s}}\right]=\left[-i\cos \varphi ,\ -i\sin \varphi ,\ i\cos \eta ,\ i\sin \eta \right]\\\left[\alpha ',\ \alpha '',\ \beta ,\ \gamma \right]=\left[i\alpha ',\ i\alpha '',\ -i\beta ,\ -i\gamma \right]\\\hline {\begin{matrix}\mathbf {(2)} {\begin{matrix}\left(\alpha -\alpha '\cos \eta -\alpha ''\sin \eta \right)^{2}=\left(\beta -\beta '\cos \eta -\beta ''\sin \eta \right)^{2}+\left(\gamma -\gamma '\cos \eta -\gamma ''\sin \eta \right)^{2}\\\left(\alpha -\beta \cos \phi -\gamma \sin \phi \right)^{2}=\left(\alpha '-\beta '\cos \phi -\gamma '\sin \phi \right)^{2}+\left(\alpha ''-\beta ''\cos \phi -\gamma ''\sin \phi \right)^{2}\\\hline {\begin{aligned}\cos \phi &={\frac {\beta -\beta '\cos \eta -\beta ''\sin \eta }{\alpha -\alpha '\cos \eta -\alpha ''\sin \eta }},&\cos \eta &={\frac {\alpha '-\beta '\cos \phi -\gamma '\sin \phi }{\alpha -\beta \cos \phi -\gamma \sin \phi }}\\\sin \phi &={\frac {\gamma -\gamma '\cos \eta -\gamma ''\sin \eta }{\alpha -\alpha '\cos \eta -\alpha ''\sin \eta }},&\sin \eta &={\frac {\alpha ''-\beta ''\cos \phi -\gamma ''\sin \phi }{\alpha -\beta \cos \phi -\gamma \sin \phi }}\end{aligned}}\end{matrix}}\\\hline \\\mathbf {(3)} {\begin{matrix}1-zz-yy={\frac {1-s's'-s''s''}{\left(\alpha -\alpha 's'-\alpha ''s''\right)^{2}}}\\\hline {\begin{aligned}y&={\frac {\beta -\beta 's'-\beta ''s''}{\alpha -\alpha 's'-\alpha ''s''}},&s'&={\frac {\alpha '-\beta 'y-\gamma 'z}{\alpha -\beta y-\gamma z}},\\z&={\frac {\gamma -\gamma 's'-\gamma ''s''}{\alpha -\alpha 's'-\alpha ''s'''}},&s''&={\frac {\alpha ''-\beta ''y-\gamma ''z}{\alpha -\beta y-\gamma z}},\end{aligned}}\end{matrix}}\end{matrix}}\left|{\begin{aligned}\alpha \alpha -\beta \beta -\gamma \gamma &=1\\\alpha '\alpha '-\beta '\beta '-\gamma '\gamma '&=-1\\\alpha ''\alpha ''-\beta ''\beta ''-\gamma ''\gamma ''&=-1\\\alpha '\alpha ''-\beta '\beta ''-\gamma '\gamma ''&=0\\\alpha ''\alpha -\beta ''\beta -\gamma ''\gamma &=0\\\alpha \alpha '-\beta \beta '-\gamma \gamma '&=0\\\\\alpha \alpha -\alpha '\alpha '-\alpha ''\alpha ''&=1\\\beta \beta -\beta '\beta '-\beta ''\beta ''&=-1\\\gamma \gamma -\gamma '\gamma '-\gamma ''\gamma ''&=-1\\\beta \gamma -\beta '\gamma '-\beta ''\gamma ''&=0\\\gamma \alpha -\gamma '\alpha '-\gamma ''\alpha ''&=0\\\alpha \beta -\alpha '\beta '-\alpha ''\beta ''&=0\end{aligned}}\right.\end{matrix}}} By setting $[\cos \phi ,\ \sin \phi ,\ \cos \eta ,\ \sin \eta ]=\left[u_{1},\ u_{2},\ u_{1}^{\prime },\ u_{2}^{\prime }\right]$ , transformation system (2) is equivalent to Lorentz transformation (1b) (n=2). Also transformation system (3) is equivalent to Lorentz transformation (1b) (n=2) up to a sign change.

Extending his previous result, Jacobi (1833) started with Cauchy's (1829) orthogonal transformation for n dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of n dimensions:[M 11]

{{\begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots +x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots +y_{n}y_{n}\\\hline \mathbf {(1)\ } {\begin{aligned}y_{\varkappa }&=\alpha _{1}^{(\varkappa )}x_{1}+\alpha _{2}^{(\varkappa )}x_{2}+\dots +\alpha _{n}^{(\varkappa )}x_{n}\\x_{\varkappa }&=\alpha _{\varkappa }^{\prime }y_{1}+\alpha _{\varkappa }^{\prime \prime }y_{2}+\dots +\alpha _{\varkappa }^{(n)}y_{n}\\\\{\frac {y_{\varkappa }}{y_{n}}}&={\frac {\alpha _{1}^{(\varkappa )}x_{1}+\alpha _{2}^{(\varkappa )}x_{2}+\dots +\alpha _{n}^{(\varkappa )}x_{n}}{\alpha _{1}^{(n)}x_{1}+\alpha _{2}^{(n)}x_{2}+\dots +\alpha _{n}^{(n)}x_{n}}}\\{\frac {x_{\varkappa }}{x_{n}}}&={\frac {\alpha _{\varkappa }^{\prime }y_{1}+\alpha _{\varkappa }^{\prime \prime }y_{2}+\dots +\alpha _{\varkappa }^{(n)}y_{n}}{\alpha _{1}^{(n)}x_{1}+\alpha _{2}^{(n)}x_{2}+\dots +\alpha _{n}^{(n)}x_{n}}}\end{aligned}}\left|{\begin{aligned}\alpha _{\varkappa }^{\prime }\alpha _{\lambda }^{\prime }+\alpha _{\varkappa }^{\prime \prime }\alpha _{\lambda }^{\prime \prime }+\dots +\alpha _{\varkappa }^{(n)}\alpha _{\lambda }^{(n)}&=0\\\alpha _{\varkappa }^{\prime }\alpha _{\varkappa }^{\prime }+\alpha _{\varkappa }^{\prime \prime }\alpha _{\varkappa }^{\prime \prime }+\dots +\alpha _{\varkappa }^{(n)}\alpha _{\varkappa }^{(n)}&=1\\\\\alpha _{1}^{(\varkappa )}\alpha _{1}^{(\lambda )}+\alpha _{2}^{(\varkappa )}\alpha _{2}^{(\lambda )}+\dots +\alpha _{n}^{(\varkappa )}\alpha _{n}^{(\lambda )}&=0\\\alpha _{1}^{(\varkappa )}\alpha _{1}^{(\varkappa )}+\alpha _{2}^{(\varkappa )}\alpha _{2}^{(\varkappa )}+\dots +\alpha _{n}^{(\varkappa )}\alpha _{n}^{(\varkappa )}&=1\end{aligned}}\right.\\\hline {\frac {x_{\varkappa }}{x_{n}}}=-i\xi _{\varkappa },\ {\frac {y_{\varkappa }}{y_{n}}}=i\nu _{\varkappa }\\1-\xi _{1}\xi _{1}-\xi _{2}\xi _{2}-\dots -\xi _{n-1}\xi _{n-1}={\frac {y_{n}y_{n}}{x_{n}x_{n}}}\left(1-\nu _{1}\nu _{1}-\nu _{2}\nu _{2}-\dots -\nu _{n-1}\nu _{n-1}\right)\\\alpha _{n}^{(\varkappa )}=i\alpha ^{(\varkappa )},\ \alpha _{\varkappa }^{(n)}=-i\alpha _{\varkappa },\ \alpha _{n}^{(n)}=\alpha \\1-\xi _{1}\xi _{1}-\xi _{2}\xi _{2}-\dots -\xi _{n-1}\xi _{n-1}={\frac {1-\nu _{1}\nu _{1}-\nu _{2}\nu _{2}-\dots -\nu _{n-1}\nu _{n-1}}{\left[\alpha -\alpha ^{\prime }\nu _{1}-\alpha ^{\prime \prime }\nu _{2}\dots -\alpha ^{(n-1)}\nu _{n-1}\right]^{2}}}\\\hline \mathbf {(2)\ } {\begin{aligned}\nu _{\varkappa }&={\frac {\alpha ^{(\varkappa )}-\alpha _{1}^{(\varkappa )}\xi _{1}-\alpha _{2}^{(\varkappa )}\xi _{2}\dots -\alpha _{n-1}^{(\varkappa )}\xi _{n-1}}{\alpha -\alpha _{1}\xi _{1}-\alpha _{2}\xi _{2}\dots -\alpha _{n-1}\xi _{n-1}}}\\\\\xi _{\varkappa }&={\frac {\alpha _{\varkappa }-\alpha _{\varkappa }^{\prime }\nu _{1}-\alpha _{2}^{\prime \prime }\nu _{2}\dots -\alpha _{\varkappa }^{(n-1)}\nu _{n-1}}{\alpha -\alpha ^{\prime }\nu _{1}-\alpha ^{\prime \prime }\nu _{2}\dots -\alpha ^{(n-1)}\nu _{n-1}}}\end{aligned}}\\\hline \xi _{1}\xi _{1}-\xi _{2}\xi _{2}-\dots -\xi _{n-1}\xi _{n-1}=1\ \Rightarrow \ \nu _{1}\nu _{1}-\nu _{2}\nu _{2}-\dots -\nu _{n-1}\nu _{n-1}=1\end{matrix}}} Transformation system (2) is equivalent to Lorentz transformation (1b) up to a sign change.

He also stated the following transformation leaving invariant the Lorentz interval:[M 12]

{\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots -u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots -w_{n-1}w_{n-1}\\\hline {{\begin{aligned}u&=\alpha w-\alpha 'w_{1}-\alpha ''w_{2}-\dots -\alpha ^{(n-1)}w_{n-1}\\u_{1}&=\alpha _{1}w-\alpha _{1}^{\prime }w_{1}-\alpha _{1}^{\prime \prime }w_{2}-\dots -\alpha _{1}^{(n-1)}w_{n-1}\\&\dots \\u_{n-1}&=\alpha _{n-1}w-\alpha _{n-1}^{\prime }w_{1}-\alpha _{n-1}^{\prime \prime }w_{2}-\dots -\alpha _{n-1}^{(n-1)}w_{n-1}\\\\w&=\alpha u-\alpha _{1}u_{1}-\alpha _{2}^{\prime \prime }u_{2}-\dots -\alpha _{n-1}u_{n-1}\\w_{1}&=\alpha 'u-\alpha _{1}^{\prime }u_{1}-\alpha _{2}^{\prime }u_{2}-\dots -\alpha _{n-1}^{\prime }u_{n-1}\\&\dots \\w_{n-1}&=\alpha ^{(n-1)}u-\alpha _{1}^{(n-1)}u_{1}-\alpha _{2}^{(n-1)}u_{2}-\dots -\alpha _{n-1}^{(n-1)}u_{n-1}\end{aligned}}\left|{\begin{aligned}\alpha \alpha -\alpha '\alpha '-\alpha ''\alpha ''\dots -\alpha ^{(n-1)}\alpha ^{(n-1)}&=+1\\\alpha _{\varkappa }\alpha _{\varkappa }-\alpha _{\varkappa }^{\prime }\alpha _{\varkappa }^{\prime }-\alpha _{\varkappa }^{\prime \prime }\alpha _{\varkappa }^{\prime \prime }\dots -\alpha _{\varkappa }^{(n-1)}\alpha _{\varkappa }^{(n-1)}&=-1\\\alpha \alpha _{\varkappa }-\alpha ^{\prime }\alpha _{\varkappa }^{\prime }-\alpha ^{\prime \prime }\alpha _{\varkappa }^{\prime \prime }\dots -\alpha ^{(n-1)}\alpha _{\varkappa }^{(n-1)}&=0\\\alpha _{\varkappa }\alpha _{\lambda }-\alpha _{\varkappa }^{\prime }\alpha _{\lambda }^{\prime }-\alpha _{\varkappa }^{\prime \prime }\alpha _{\lambda }^{\prime \prime }\dots -\alpha _{\varkappa }^{(n-1)}\alpha _{\lambda }^{(n-1)}&=0\\\\\alpha \alpha -\alpha _{1}\alpha _{1}-\alpha _{2}\alpha _{2}\dots -\alpha _{n-1}\alpha _{n-1}&=+1\\\alpha _{\varkappa }\alpha _{\varkappa }-\alpha _{1}^{\varkappa }\alpha _{1}^{\varkappa }-\alpha _{2}^{\prime \prime }\alpha _{2}^{\prime \prime }\dots -\alpha _{n-1}^{(\varkappa )}\alpha _{n-1}^{(\varkappa )}&=-1\\\alpha \alpha ^{(\varkappa )}-\alpha _{1}\alpha _{1}^{(\varkappa )}-\alpha _{2}\alpha _{2}^{(\varkappa )}\dots -\alpha _{n-1}\alpha _{n-1}^{(\varkappa )}&=0\\\alpha ^{(\varkappa )}\alpha ^{(\lambda )}-\alpha _{1}^{(\varkappa )}\alpha _{1}^{\lambda l)}-\alpha _{2}^{(\varkappa )}\alpha _{2}^{(\lambda )}\dots -\alpha _{n-1}^{(\varkappa )}\alpha _{n-1}^{(\lambda )}&=0\end{aligned}}{\text{ }}\right.}\end{matrix}} This is equivalent to Lorentz transformation (1a) up to a sign change.

### Lebesgue (1837) – Homogeneous coordinates

w:Victor-Amédée Lebesgue (1837) summarized the previous work of Gauss (1818), Jacobi (1827, 1833), Cauchy (1829). He started with the orthogonal transformation[M 13]

{\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots +x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n}^{2}\ (9)\\\hline {{\begin{aligned}x_{1}&=a_{1,1}y_{1}+a_{1,2}y_{2}+\dots +a_{1,n}y_{n}\\x_{2}&=a_{2,1}y_{1}+a_{2,2}y_{2}+\dots +a_{2,n}y_{n}\\\dots \\x_{n}&=a_{n,1}x_{1}+a_{n,2}x_{2}+\dots +a_{n,n}x_{n}\\\\y_{1}&=a_{1,1}x_{1}+a_{2,1}x_{2}+\dots +a_{n,1}x_{n}\\y_{2}&=a_{1,2}x_{1}+a_{2,2}x_{2}+\dots +a_{n,2}x_{n}\ (12)\ \\\dots \\y_{n}&=a_{1,n}x_{1}+a_{2,n}x_{2}+\dots +a_{n,n}x_{n}\end{aligned}}\left|{\begin{aligned}a_{1,\alpha }^{2}+a_{2,\alpha }^{2}+\dots +a_{n,\alpha }^{2}&=1&(10)\\a_{1,\alpha }a_{1,\beta }+a_{2,\alpha }a_{2,\beta }+\dots +a_{n,\alpha }a_{n,\beta }&=0&(11)\\a_{\alpha ,1}^{2}+a_{\alpha ,2}^{2}+\dots +a_{\alpha ,n}^{2}&=1&(13)\\a_{\alpha ,1}a_{\beta ,1}+a_{\alpha ,2}a_{\beta ,2}+\dots +a_{\alpha ,n}a_{\beta ,n}&=0&(14)\end{aligned}}\right.}\end{matrix}} In order to achieve the invariance of the Lorentz interval[M 14]

$x_{1}^{2}+x_{2}^{2}+\dots +x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n-1}^{2}-y_{n}^{2}$ he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first n-1 equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=n change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the n-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the n-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign.

These instructions give Lorentz transformation (1a) in the form:
{{\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots +x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n-1}^{2}-y_{n}^{2}\\\hline {\begin{aligned}x_{1}&=a_{1,1}y_{1}+a_{1,2}y_{2}+\dots +a_{1,n}y_{n}\\x_{2}&=a_{2,1}y_{1}+a_{2,2}y_{2}+\dots +a_{2,n}y_{n}\\\dots \\x_{n}&=a_{n,1}x_{1}+a_{n,2}x_{2}+\dots +a_{n,n}x_{n}\\\\y_{1}&=a_{1,1}x_{1}+a_{2,1}x_{2}+\dots +a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\y_{2}&=a_{1,2}x_{1}+a_{2,2}x_{2}+\dots +a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\\dots \\-y_{n}&=a_{1,n}x_{1}+a_{2,n}x_{2}+\dots +a_{n-1,n}x_{n-1}-a_{n,n}x_{n}\end{aligned}}\left|{\begin{aligned}a_{1,\alpha }^{2}+a_{2,\alpha }^{2}+\dots +a_{n-1,\alpha }^{2}-a_{n,\alpha }^{2}&=1\\a_{1,n}^{2}+a_{2,n}^{2}+\dots +a_{n-1,n}^{2}-a_{n,n}^{2}&=-1\\a_{1,\alpha }a_{1,\beta }+a_{2,\alpha }a_{2,\beta }+\dots +a_{n-1,\alpha }a_{n-1,\beta }-a_{n,\alpha }a_{n,\beta }&=0\\a_{\alpha ,1}^{2}+a_{\alpha ,2}^{2}+\dots +a_{\alpha ,n-1}^{2}-a_{\alpha ,n}^{2}&=1\\a_{n,1}^{2}+a_{n,2}^{2}+\dots +a_{n,n-1}^{2}-a_{n,n}^{2}&=-1\\a_{\alpha ,1}a_{\beta ,1}+a_{\alpha ,2}a_{\beta ,2}+\dots +a_{\alpha ,n-1}a_{\beta ,n-1}-a_{\alpha ,n}a_{\beta ,n}&=0\end{aligned}}\right.\end{matrix}}} He went on to redefine the variables of the Lorentz interval and its transformation:[M 15]

{\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots +x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots +y_{n-1}^{2}-y_{n}^{2}\\\downarrow \\{\begin{aligned}x_{1}&=x_{n}\cos \theta _{1},&x_{2}&=x_{n}\cos \theta _{2},\dots &x_{n-1}&=x_{n}\cos \theta _{n-1}\\y_{1}&=y_{n}\cos \phi _{1},&y_{2}&=y_{n}\cos \phi _{2},\dots &y_{n-1}&=y_{n}\cos \phi _{n-1}\end{aligned}}\\\downarrow \\\cos ^{2}\theta _{1}+\cos ^{2}\theta _{2}+\dots +\cos ^{2}\theta _{n-1}=1\\\cos ^{2}\phi _{1}+\cos ^{2}\phi _{2}+\dots +\cos ^{2}\phi _{n-1}=1\\\hline \\\cos \theta _{i}={\frac {a_{i,1}\cos \phi _{1}+a_{i,2}\cos \phi _{2}+\dots +a_{i,n-1}\cos \phi _{n-1}+a_{i,n}}{a_{n,1}\cos \phi _{1}+a_{n,2}\cos \phi _{2}+\dots +a_{n,n-1}\cos \phi _{n-1}+a_{n,n}}}\\(i=1,2,3\dots n)\end{matrix}} Setting $[\cos \theta _{i},\ \cos \phi _{i}]=\left[u_{s},\ u_{s}^{\prime }\right]$ it is equivalent to Lorentz transformation (1b).

### Bour (1856) – Homogeneous coordinates

Following Gauss (1818), w:Edmond Bour (1856) wrote the transformations:[M 16]

{\begin{matrix}\cos ^{2}E+\sin ^{2}E-1=k\left(\cos ^{2}T+\sin ^{2}T-1\right)\\\hline \left.{\begin{matrix}\mathbf {(1)} \ {\begin{aligned}\cos E&={\frac {\alpha +\alpha '\cos T+\alpha ''\sin T}{\gamma +\gamma '\cos T+\gamma ''\sin T}}\\\sin E&={\frac {\beta +\beta '\cos T+\beta ''\sin T}{\gamma +\gamma '\cos T+\gamma ''\sin T}}\end{aligned}}\\\hline \\k=+1\\t=\gamma +\gamma '\cos T+\gamma ''\sin T,\\1=u,\ \cos T=u',\ \sin T=u',\\t=z,\ t\cos E=x,\ t\sin E=y\\\downarrow \\\mathbf {(2)} {\begin{aligned}x&=\alpha u+\alpha 'u'+\alpha ''u''\\y&=\beta u+\beta 'u'+\beta ''u''\\z&=\gamma u+\gamma 'u'+\gamma ''u''\\\\u&=\gamma z-\alpha x-\beta y\\u'&=\alpha 'x+\beta 'y'-\gamma 'z\\u''&=\alpha ''x+\beta ''y-\gamma ''z\end{aligned}}\end{matrix}}\right|{{\begin{aligned}-\alpha ^{2}-\beta ^{2}+\gamma ^{2}&=k\\-\alpha ^{\prime 2}-\beta ^{\prime 2}+\gamma ^{\prime 2}&=-k\\-\alpha ^{\prime \prime 2}-\beta ^{\prime \prime 2}+\gamma ^{\prime \prime 2}&=-k\\\alpha \alpha '+\beta \beta '-\gamma \gamma '&=0\\\alpha \alpha ''+\beta \beta ''-\gamma \gamma ''&=0\\\alpha '\alpha ''+\beta '\beta ''-\gamma '\gamma ''&=0\\\\\alpha ^{2}-\alpha ^{\prime 2}-\alpha ^{\prime \prime 2}&=-k\\\beta ^{2}-\beta ^{\prime 2}-\beta ^{\prime \prime 2}&=-k\\\gamma ^{2}-\gamma ^{\prime 2}-\gamma ^{\prime \prime 2}&=k\\\beta \gamma -\beta '\gamma '-\beta ''\gamma ''&=0\\\alpha \gamma -\alpha '\gamma '-\alpha ''\gamma ''&=0\\\alpha \beta -\alpha '\beta '-\alpha ''\beta ''&=0\end{aligned}}}\end{matrix}} Setting $[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime },u_{2}^{\prime }\right]$ , the transformation system (2) becomes Lorentz transformation (1b) (n=2). Transformation system (2) is equivalent to Lorentz transformation (1a) (n=2), implying $x^{2}+y^{2}-z^{2}=u^{\prime 2}+u^{\prime \prime 2}-u^{2}$ ### Somov (1863) – Homogeneous coordinates

Following Gauss (1818), Jacobi (1827, 1833), and Bour (1856), w:Osip Ivanovich Somov (1863) wrote the transformation systems:[M 17]

{\begin{matrix}\left.{\begin{aligned}\cos \phi &={\frac {m\cos \psi +n\sin \psi +s}{m''\cos \psi +n''\sin \psi +s''}}\\\sin \phi &={\frac {m'\cos \psi +n'\sin \psi +s'}{m''\cos \psi +n''\sin \psi +s''}}\end{aligned}}\right|{\begin{matrix}\cos ^{2}\phi +\cos ^{2}\phi =1\\\cos ^{2}\psi +\cos ^{2}\psi =1\end{matrix}}\\\hline \mathbf {(1)} \ {\begin{aligned}\cos \phi &=x,&\cos \psi &=x'\\\sin \phi &=y,&\sin \psi &=y'\end{aligned}}\ \left|{\begin{aligned}x&={\frac {mx'+ny'+s}{m''x'+n''y'+s''}}\\y&={\frac {m'x'+n'y'+s'}{m''x'+n''y'+s''}}\end{aligned}}\right|\ {\begin{matrix}x^{2}+y^{2}=1\\x^{\prime 2}+y^{\prime 2}=1\end{matrix}}\\\hline {\begin{aligned}\cos \phi &={\frac {x}{z}},&\cos \psi &={\frac {x'}{z'}}\\\sin \phi &={\frac {y}{z}},&\sin \psi &={\frac {y'}{z'}}\end{aligned}}\ \left|{\begin{aligned}{\frac {x}{z}}&={\frac {mx'+ny'+sz'}{m''x'+n''y'+s''z'}}\\{\frac {y}{z}}&={\frac {m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'}}\end{aligned}}\right|\ {\begin{matrix}x^{2}+y^{2}=z^{2}\\x^{\prime 2}+y^{\prime 2}=z^{\prime 2}\end{matrix}}\\\hline \mathbf {(2)} \ \left.{\begin{aligned}x&=mx'+ny'+sz'\\y&=m'x'+n'y'+s'z'\\z&=m''x'+n''y'+s''z'\\\\x'&=mx+m'y-m''z\\y'&=nx+n'y-n''z\\z'&=-sx-s'y+s''z\\\\dx&=mdx'+ndy'+sdz'\\dy&=m'dx'+n'dy'+s'dz'\\dz&=m''dx'+n''dy'+s''dz'\end{aligned}}\right|{{\begin{aligned}m^{2}+m^{\prime 2}-m^{\prime \prime 2}&=1\\n^{2}+n^{\prime 2}-n^{\prime \prime 2}&=1\\-s^{2}-s^{\prime 2}+s^{\prime \prime 2}&=1\\ns+n's'-n''s''&=0\\sm+s'm'-s''m''&=0\\mn+m'n'-m''n''&=0\\\\m^{2}+n^{2}-s^{2}&=1\\m^{\prime 2}+n^{\prime 2}-s^{\prime 2}&=1\\-m^{\prime \prime 2}-n^{\prime \prime 2}+s^{\prime \prime 2}&=1\\-m'm''-n'n''+s's''&=0\\-m''m-n''n+s''s&=0\\mm'+nn'-ss'&=0\end{aligned}}}\\dx^{2}+dy^{2}-dz^{2}=dx^{\prime 2}+dy^{\prime 2}-dz^{\prime 2}\end{matrix}} Transformation system (1) is equivalent to Lorentz transformation (1b) (n=2). Transformation system (2) is equivalent to Lorentz transformation (1a) (n=2).

### Killing (1878–1893)

#### Weierstrass coordinates

w:Wilhelm Killing (1878–1880) described non-Euclidean geometry by using Weierstrass coordinates (named after w:Karl Weierstrass who described them in lectures in 1872 which Killing attended) obeying the form

$k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}$ [M 18] with $ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}$ [M 19]

or[M 20]

$k^{2}x_{0}^{2}+x_{1}^{2}+\dots +x_{n}^{2}=k^{2}$ where k is the reciprocal measure of curvature, $k^{2}=\infty$ denotes w:Euclidean geometry, $k^{2}>0$ w:elliptic geometry, and $k^{2}<0$ hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form.[M 21] In (1879/80) he wrote the corresponding transformations as a general rotation matrix[M 22]

${\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\\hline {\begin{matrix}\cos \eta \tau +\lambda ^{2}{\frac {1-\cos \eta \tau }{\eta ^{2}}},&\nu {\frac {\sin \eta \tau }{\eta }}+\lambda \mu {\frac {1-\cos \eta \tau }{\eta ^{2}}},&-\mu \sin {\frac {\eta \tau }{\eta }}+\nu \lambda {\frac {1-\cos \eta \tau }{\eta ^{2}}}\\-k^{2}\nu {\frac {\sin \eta \tau }{\eta }}+k^{2}\lambda \mu {\frac {1-\cos \eta \tau }{\eta ^{2}}},&\cos \eta \tau +\mu ^{2}{\frac {1-\cos \eta \tau }{\eta ^{2}}},&\lambda {\frac {\sin \eta \tau }{\eta }}+k^{2}\mu \nu {\frac {1-\cos \eta \tau }{\eta ^{2}}}\\k^{2}\mu {\frac {\sin \eta \tau }{\eta }}+k^{2}\nu \lambda {\frac {1-\cos \eta \tau }{\eta ^{2}}},&-\lambda {\frac {\sin \eta \tau }{\eta }}+k^{2}\mu \nu {\frac {1-\cos \eta \tau }{\eta ^{2}}},&\cos \eta \tau +\nu ^{2}{\frac {1-\cos \eta \tau }{\eta ^{2}}}\end{matrix}}\\\left(\lambda ^{2}+k^{2}\mu ^{2}+k^{2}\nu ^{2}=\eta ^{2}\right)\end{matrix}}$ In (1885) he wrote the Weierstrass coordinates and their transformation as follows:[M 23]

{\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime 2}+x^{\prime 2}+y^{\prime 2}\\ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\\hline {\begin{aligned}k^{2}p'&=k^{2}wp+w'x+w''y\\x'&=ap+a'x+a''y\\y'&=bp+b'x+b''y\\\\k^{2}p&=k^{2}wp'+ax'+by'\\x&=w'p'+a'x+b'y'\\y&=w''p'+a''x'+b''y'\end{aligned}}\left|{{\begin{aligned}k^{2}w^{2}+w^{\prime 2}+w^{\prime \prime 2}&=k^{2}\\{\frac {a^{2}}{k^{2}}}+a^{\prime 2}+a^{\prime \prime 2}&=1\\{\frac {b^{2}}{k^{2}}}+b^{\prime 2}+b^{\prime \prime 2}&=1\\aw+a'w'+a''w''&=0\\bw+b'w'+b''w''&=0\\{\frac {ab}{k^{2}}}+a'b'+a''b''&=0\\\\k^{2}w^{2}+a^{2}+b^{2}&=k^{2}\\{\frac {w^{\prime 2}}{k^{2}}}+a^{\prime 2}+b^{\prime 2}&=1\\{\frac {w^{\prime \prime 2}}{k^{2}}}+a^{\prime \prime 2}+b^{\prime \prime 2}&=1\\ww'+aa'+bb'&=0\\ww''+aa''+bb''&=0\\{\frac {w'w''}{k^{2}}}+a'a''+b'b''&=0\end{aligned}}}\right.\end{matrix}} This is similar to Lorentz transformation (1a) (n=2) with $k^{2}=-1$ In (1885) he also gave the transformation for n dimensions:[M 24]

{\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots +x_{n}^{2}=k^{2}\\ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots +dx_{n}^{2}\\\hline \left.{\begin{aligned}k^{2}\xi _{0}&=k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots +a_{0n}x_{0}\\\xi _{\varkappa }&=a_{\varkappa 0}x_{0}+a_{\varkappa 1}x_{1}+\dots +a_{\varkappa n}x_{n}\\\\k^{2}x_{0}&=a_{00}k^{2}\xi _{0}+a_{10}\xi _{1}+\dots +a_{n0}\xi _{n}\\x_{\varkappa }&=a_{0\varkappa }\xi _{0}+a_{1\varkappa }\xi _{1}+\dots +a_{n\varkappa }\xi _{n}\end{aligned}}\right|{{\begin{aligned}k^{2}a_{00}^{2}+a_{10}^{2}+\dots +a_{n0}^{2}&=k^{2}\\a_{00}a_{0\varkappa }+a_{10}a_{1\varkappa }+\dots +a_{n0}a_{n\varkappa }&=0\\{\frac {a_{0\iota }a_{0\varkappa }}{k^{2}}}+a_{0\iota }a_{1\varkappa }+\dots +a_{n\iota }a_{n\varkappa }=\delta _{\iota \kappa }&=1\ (\iota =\kappa )\ {\text{or}}\ 0\ (\iota \neq \kappa )\end{aligned}}}\end{matrix}} This is similar to Lorentz transformation (1a) with $k^{2}=-1$ In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force.[M 25] Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting $k^{2}=-1$ and using p,x,y as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid $p^{2}-x^{2}-y^{2}=1$ (known as w:hyperboloid model),[M 26] by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space. Finally, in (1893) he wrote:[M 27]

{\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\\hline {\begin{aligned}t'&=at+bu+cv\\u'&=a't+b'u+c'v\\v'&=a''t+b''u+c''v\end{aligned}}\left|{\begin{aligned}k^{2}a^{2}+a^{\prime 2}+a^{\prime \prime 2}&=k^{2}\\k^{2}b^{2}+b^{\prime 2}+b^{\prime \prime 2}&=1\\k^{2}c^{2}+b^{\prime 2}+c^{\prime \prime 2}&=1\\k^{2}ab+a'b'+a''b''&=0\\k^{2}ac+a'c'+a''c''&=0\\k^{2}bc+b'c'+b''c''&=0\end{aligned}}\right.\end{matrix}} This is equivalent to Lorentz transformation (1a) (n=2) with $k^{2}=-1$ and for n dimensions[M 28]