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Lorentz's equations
Let's consider two entities E and E ' that move with speeds different.
Let's suppose, besides, that their system of reference : (x, z, y, t) and (x ', y', z ', t ') are such that the axis x coincides with the axis x '.
When an information vector of speed V (for example, a photon of the light) is moving within E, in a parallel direction to the axis x , it covers a distance : x = Vt ; thus x - Vt = 0.
In E ', if this vector has the same speed V, in a similar way we shall have: x ' = Vt ' and so, x ' - Vt ' = 0.
What has been Lorentz's reasoning ?
If the information vectors are the light's constituents : the photons of speed c, postulated constant in space (299,792,458 m/s ), and if propagation of this light is according to positive direction of the axis x ,
V = c and the previous formulae become :
(1) x - ct = x' - ct' = 0 = (a + b) (x - ct) = ax + bx - act - bct.
On the other hand, if this propagation is developping in negative direction of the axis x :
(2) x + ct = x' + ct' = 0 = (a - b) (x + ct) = ax - bx + act – bct,
a and b being two constants which facilitate the calculations.
After adding of the equations (1) and (2) we obtain :
(5) x' = ax - bct,
and by subtraction:
(6) ct' = act - bx.
Thus, by construction and for the origin of the axis x', knowing that x' = 0 :
(7) ax = bct
So, if the origin of the axis x', has a speed v in relation to E, the axiss x and x ' being parallel, this relative speed of E ' in relation to E, expressed by means of the equation (7 ), will be such that :
(8) av = bc.
Then, for t' = 0 which corresponds to the instantaneousness in E', considering (8) and after elimination of t in the formulations (5) and (6), it appears that :
(9) 
and for two points of the axis x, distant of "d" :
According the previous equation x ' = ax - bct, when t = 0 in the instantaneousness,
x' will be equal to ax, and D x associated to length's perception "d" measured in E' (a precise physical length) will be : D x = d/a .
Obviously, this distance "d" is the same whatever is reference system, D x must thus be equal to D x' ; consequently :
 or : 
And as : av = bc (8),
two expressions of the perceptions's interaction of a same phenomenon in two reference systems having a relative speed, will be represented by the following equations :
(10)  and (11)  ,
said, Lorentz's transformation equations.
More exactly, length of a standard meter which has a speed v in the system E, in a parallel direction with the axis x, will not be for the motionless observer, of 1 metre but of : metre.
In a similar way, the length measurement of a standard motionless meter by an observer who has a speed v, will depend on same relativity coefficient : .
As for the durations (for the time),
speed, lapse of time and length "l" being united by the simple formula l = vt,
their measurement too, will depend of a relativity coefficient of same type. So, the lapses of time punctuated by a clock which has a speed v, appear longer for a motionless observer, corrective coefficient being : 
These equations were first expressions of the relative character of the light's perception.
Let's insist more, so much in research of primordial causes, rigour in speech is imperative.
More exactly, these expressions concern relative character of the perceptions's interpretation of light (of light-waves),
knowing that any interpretation necessitates transcendent order faculties,
notably faculties that enable recognitions, judgments, selections and theories.
Evidently, these equations fascinated the scientists, in particular, Henri Poincaré (1854 - 1912) ; he anticipated existence, probably the first, of a relation between energy, mass and light’s speed (cf. Conference of St Louis - U. S - in 1904).
Nevertheless, why A. Einstein has been the one who expressed a quintessence of Lorentz's works ?
He elaborated thus his special theory of relativity (published in November, 1905), a theory rather easy for one who has a small mathematical culture.
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