A common topic in ontologies of Crusius’ time, not that usual in modern ontologies, is quantities - back then, general philosophers were keen to explain what mathematics is all about, while nowadays this question is more and more left for special branch called philosophy of mathematics. Crusius follows the tradition and starts by defining quantity as such a property of a thing, by which something is posited more than once.
Crusius notes that at least complex concrete things naturally have a quantity - they consist of many things. Furthermore, even simple concrete things have quantifiable features - they have forces, and even though they are indivisible, they still are spatial and thus have some magnitude. Then again, some abstractions are not quantifiable, Crusius says: there are no levels of existence, but all existing things exist as much as others. Crusius also notes in passing the possibility of infinite quantities, but at once declares that we finite beings cannot really know anything about them.
Quantities come in different types, Crusius continues, for instance, quantity of a force differs from a quantity of an extension. The difference between these types becomes important, when we start to measure the quantities. Measuring, Crusius says, involves determining a relation of a quantity to some known quantity. As such, this kind of comparison is possible only between quantities of the same type (there’s no sense in measuring weight with a ruler). Still, Crusius admits, quantities of different type can be compared indirectly. Firstly, we can compare them through relations of quantities - for instance, we can say that punishments should be proportional to the crimes punished. Secondly, the comparison can be done through causal links, for example, the resistance of a body can be compared with the striving of a soul, because one has the effect of hindering the other.
To determine a quantity perfectly, Crusius says, we must represent its parts distinctly. This requires expressing the quantity as a number of distinctly thought units. These units might be naturally distinct - for instance, when we count things distinguished by natural limits, like cows - or arbitrarily chosen, for example, when we compare length of a thing to a measuring stick. Since a given quantity might not be expressible as a number of arbitrarily chosen units, Crusius also introduces fractions (no mention of irrational numbers, though).
An extreme case of natural units, for Crusius, is naturally provided by simple substances. Crusius admits that measuring complex substances by their simple parts is impossible, since we do not perceive these ultimate constituents. Still, he continues, understanding the nature of these simple parts can help us in picking suitable units for measurement: for instance, when we note that movement should be ideally measured by checking how many simple substances move through smallest measures of space, we can surmise that movement could be measured by checking how many things move through a certain space.
Crusius spends the majority of the rest of the chapter discussing a hotly debated topic of the time, namely, the so-called question of living forces. The point of the debate, at least as conceived by Crusius, is how to measure the quantity of an action, such as movement. Crusius’ take is that while abstractly taken this quantity can be expressed as a multiple of the strength of the action (in case of movement, mass of the moving object) and its velocity, we must also account for the resistance encountered by the action and thus use the square of velocity to determine the action.
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