This question tests your understanding of verb patterns with "supposed to" and perfect infinitives.

The correct structure is: be supposed + to + infinitive. This expresses what someone is believed to have done or what is expected to happen.

When we talk about a past action that someone is believed to have done, we use the perfect infinitive ($\mathrm{to} \mathrm{have} \mathrm{invented}$). The perfect infinitive is formed with $\mathrm{to} \mathrm{have} + \mathrm{past} \mathrm{participle}$.

Let's analyze the sentence: "Galileo is supposed _____ the telescope."

Step 1: Identify the main structure. The verb is "is supposed", which requires "to" + an infinitive form. This immediately eliminates the options "have invented" and "having invented".

Step 2: Choose between the remaining infinitives: "to invent" and "to have invented". The sentence is about the historical belief that Galileo performed the action of inventing the telescope in the past. Therefore, we need the perfect infinitive ($\mathrm{to} \mathrm{have} \mathrm{invented}$) to indicate an action completed prior to the present time.

Final Answer: The most appropriate word is to have invented. The complete sentence is: "Galileo is supposed to have invented the telescope."

Related Concepts & Formulae

Verb + to + infinitive: Many verbs and adjectives are followed by 'to' and the base form of a verb (infinitive). Common examples include: supposed to, likely to, certain to, eager to, etc.

Perfect Infinitive (to have + past participle): This structure is used after verbs, adjectives, and nouns to refer to an action that happened before the time of the main verb. Formula: $\mathrm{to} \mathrm{have} + \mathrm{V3} (\mathrm{past} \mathrm{participle})$.

Example: She claims to have seen a UFO. (The seeing happened before the claiming).