  Mechanics studies the relationships between space, time, and matter, which
can be expressed in terms of the dimensions of length $\mathcal{L}$, time
$\mathcal{T}$, and mass $\mathcal{M}$. Each dimension broadens the scope of
mechanics, from geometric quantities with dimensions of the form
$\mathcal{L}^x$ (like lengths or areas), to kinematic quantities of the form
$\mathcal{L}^x\mathcal{T}^y$ (like speeds or accelerations), and eventually
``mass-carrying'' quantities such as mass, force, momentum, energy, action,
power, viscosity, etc. These standard mechanical quantities have dimensions of
the form $\mathcal{M}\mathcal{L}^x\mathcal{T}^y$, where $x$ and $y$ are
integers. In this contribution, we use this dimensional structure to arrange
these mass-carrying quantities into a table indexed by $x$ and $y$. Ratios of
quantities in the same rows provide characteristic lengths, and in the same
columns characteristic times, encompassing a great variety of physical
phenomena from atomic to astronomical scales. Most generally, we show that
picking duos of mechanical quantities that are neither on the same row nor
column yields dynamics, where one mechanical quantity is understood as
impelling motion, while the other is impeding it. The force and the mass are
the prototypes of impelling and impeding factors, but many other duos are
possible. This review provides a novel synthesis revealing the power of
dimensional analysis, to understand processes governed by the interplay of two
mechanical quantities. This elementary decomposition of space, time and motion
into pairs of mechanical factors is the foundation of ``dimensional
mechanics'', a method that this review wishes to promote and advance. The
review is complemented by online video lectures, which initiate a discussion on
the elaborate interplay of two or more mechanical quantities.
