Ernst Mach devotes 7 pages to Lambert in The Principles of Physical Optics. Lambert's basic assumptions were supported by experiments involving the visual comparison of illuminated surfaces. He wrote in Latin near the end of the Enlightenment which makes him difficult for modern readers to understand.
Here is some of what he actually said in his Photometria:
"Diximus vero, quod & vulgo notissimum.
1°. Duas pluresue candelas plus illuminare quam unica.
2°. Objectum lumini propius admotum clarius fieri.
3°. Lumen oblique incidens in superficiem, eam minus illuminare."
"We have said, however, what is commonly well known.
1°. Two or more candles illuminate more than one.
2°. The object is made brighter as the light moves nearer.
3°. Light obliquely incident on a surface, illuminates it less."
Sin of inclination factor
"Vis illuminans simulque ipsa illuminatio decrescat in ratione sinus anguli emanations."
"The illuminating power diminishes as the illumination itself in proportion to the sine of the angle of the emanations."
Inverse square law
"Si corpus luminosum, fuerit sphaericum, illuminatio absoluta in A se habebit ad quamlibet aliam normalem, in E, ut se havet quadratum distantiae CE ad quadratum semidiametri corporis CA, adeoque reciproce in ratione duplicata distantiae objecti E a centro corporis C."
"If a luminous body is spherical, it will have absolute illumination at A to that of any normal at E, as it has the square of the distance CE to the square of the radius of the body CA, and so in proportion to the inverse square of the distance E from the center of the body C."
Calculation of the emission into a hemisphere
Note the use of calculus.
The translations are mine with the help of Google translate.