What is the name for the general relation of the volume of an $n$-simplex shown in $(\star\star)$ below? In which specific books I can find the formula?
Some words about the meaning of equation $(\star\star)$
The general relation $(\star\star)$ is derived from $(\star)$. Special cases of $(\star)$ are equations (1-3), i.e. the length of a line segment $(n=1)$ from its base point, the area of a triangle $(n=2)$ from its base length, and the volume of a tetrahedron $(n=3)$ from its base area:
$$\begin{align} \text{line length}\;&=\;\frac{\text{1 [Point]} \times \text{height}}{1!} \tag{1}\\[6pt] \text{triangle area}\;&=\;\frac{\text{base length} \times \text{height}}{2!} \tag{2}\\[6pt] \text{tetrahedron volume}\;&=\;\frac{\text{base area} \times \text{height}}{3!} \tag{3} \end{align}$$
The formula $(\star)$ below gives the $n$-volumes for any $n$ using its $(n-1)$-volume and the height. The formula $(\star\star)$ gives the $n$-volume without knowing the $(n-1)$-volume. The proof can be found in this Math.SE answer. What is the name of the relation $(\star\star)$ and in which book it is derived?
If $\ V_n\big(v_0,v_1,\dots,v_n\big)\ $ is the $n$-volume of the $n$-simplex $\ \mathcal{S}\ $with vertices $\ v_0,v_1,\dots,v_n\ $, then $$ V_n\big(v_0,v_1,\dots,v_n\big)=\frac{V_{n-1}\big(v_0,v_1,\dots,v_{n-1}\big)\big\|\big(I-P_{n-1}\big)\big(v_n-v_0\big)\big\|}{n}\ , \tag{$\star$}$$ where $\ P_n\ $ is the perpendicular projection onto the $\ n-1$-dimensional space spanned by $\ v_{n-1}-v_0,$$v_{n-1}-v_0,$$\dots, v_1-v_0\ $. The factor $\ \big\|\big(I-P_{n-1}\big)\big(v_n-v_0\big)\big\|\ $ in this product is just the height of the vertex $\ v_n\ $ above the $n-1$-simplex with vertices $\ v_0,v_1,\dots,v_{n-1}\ $, taken as a base of $\ \mathcal{S}\ $. By induction, therefore, $$ V_n\big(v_0,v_1,\dots,v_n\big)=\frac{1}{n!}\|v_1-v_0\|\prod_{j=1}^{n-1}\big\|\big(I-P_j\big)\big(v_{j+1}-v_0\big)\big\|\ \tag{$\star\star$}$$