Adam Clarke on Matthew 16:24

Will come after me - i.e. to be my disciple. This discourse was intended to show Peter and the rest of the disciples the nature of his kingdom; and that the honor that cometh from the world was not to be expected by those who followed Christ.

The principles of the Christian life are:

First. To have a sincere desire to belong to Christ - If any man be Willing to be my disciple, etc.

Secondly. To renounce self-dependence, and selfish pursuits - Let him deny Himself.

Thirdly. To embrace the condition which God has appointed, and bear the troubles and difficulties he may meet with in walking the Christian road - Let him take up His Cross.

Fourthly. To imitate Jesus, and do and suffer all in his spirit - Let him Follow Me.

Let him deny himself - Απαρνησασθω may well be interpreted, Let him deny, or renounce, himself fully - in all respects - perseveringly. It is a compounded word, and the preposition απο abundantly increases the meaning. A follower of Christ will need to observe it in its utmost latitude of meaning, in order to be happy here, and glorious hereafter. A man's self is to him the prime cause of most of his miseries. See the note on Mar 8:34.