**Shelter** **himself** **under** **the** **authority** **of** **Plato**; **but**, **as** **the** **Xenocratean** **numbers**, **though** **professedly** **ideal** **as** **well** **as** **mathematical**, **were** **in** **fact** **mathematical** **only**, **this** **return** **to** **the** **Platonic** **terminology** **was** **no** **more** **than** **an** **empty** **form**.

**In** **thus** **reverting** **to** **the** **crudities** **of** **certain** **Pythagoreans**, **he** **laid** **himself** **open** **to** **the** **criticisms** **of** **Aristotle**, **who**, **in** **his** **Metaphysics**, **recognizing** **amongst** **contemporary** **Platonists** **three** **principal** **groups** - (1) **those** **who**, **like** **Plato**, **distinguished** **mathematical** **and** **ideal** **numbers**; (2) **those** **who**, **like** **Xenocrates**, **identified** **them**; **and** (3) **those** **who**, **like** **Speusippus**, **postulated** **mathematical** **numbers** **only** - **has** **much** **to** **say** **against** **the** **Xenocratean** **interpretation** **of** **the** **theory**, **and** **in** **particular** **points** **out** **that**, **if** **the** **ideas** **are** **numbers** **made** **up** **of** **arithmetical** **units**, **they** **not** **only** **cease** **to** **be** **principles**, **but** **also** **become** **subject** **to** **arithmetical** **operations**.