Definition and Context:
The concept of "Public Appearance to Israel" refers to the instances in the Bible where God, His angels, or His chosen messengers make themselves known to the nation of Israel in a visible and often miraculous manner. These appearances serve various purposes, including delivering messages, providing guidance, or demonstrating divine power and authority.

Old Testament Instances:

1. Theophanies:
The Old Testament contains several theophanies, where God appears in a visible form. One of the most significant is the appearance of God to Moses in the burning bush (Exodus 3:2-6). Here, God reveals His name, Yahweh, and commissions Moses to lead the Israelites out of Egypt.

2. Mount Sinai:
At Mount Sinai, God makes a dramatic public appearance to the Israelites. The mountain is enveloped in smoke, and there are thunder, lightning, and the sound of a trumpet (Exodus 19:16-19). This event underscores the giving of the Law and establishes the covenant between God and Israel.

3. The Angel of the Lord:
The Angel of the Lord appears multiple times, often as a divine messenger. In Judges 6:11-24, the Angel of the Lord appears to Gideon, calling him to deliver Israel from the Midianites. These appearances often carry significant messages or instructions for Israel.

4. Prophetic Visions:
Prophets like Isaiah and Ezekiel receive visions that are meant for the entire nation. Isaiah's vision of the Lord in the temple (Isaiah 6:1-8) and Ezekiel's vision of the glory of God (Ezekiel 1:1-28) are examples where the prophets are given messages to convey to Israel.

New Testament Instances:

1. The Incarnation of Christ:
The ultimate public appearance to Israel is the incarnation of Jesus Christ. As the Word made flesh (John 1:14), Jesus' life and ministry are a direct revelation of God to His people. His teachings, miracles, and fulfillment of prophecy are a testament to His divine nature and mission.

2. The Transfiguration:
Jesus' transfiguration is a significant public appearance to select disciples, revealing His divine glory. In Matthew 17:1-8, Jesus is transfigured before Peter, James, and John, with Moses and Elijah appearing alongside Him. This event confirms Jesus' messianic identity and divine authority.

3. Post-Resurrection Appearances:
After His resurrection, Jesus appears to many, including His disciples and over five hundred others (1 Corinthians 15:6). These appearances serve to confirm His victory over death and His identity as the risen Lord.

4. Pentecost:
The outpouring of the Holy Spirit at Pentecost is a public manifestation of God's presence and power. Acts 2:1-4 describes how the Holy Spirit descends on the apostles, enabling them to speak in various tongues and proclaim the gospel to people from different nations gathered in Jerusalem.

Significance:
Public appearances to Israel are pivotal moments in biblical history, serving to affirm God's covenant, reveal His will, and demonstrate His power and presence among His people. These events often mark significant transitions or affirmations of God's ongoing relationship with Israel and, by extension, His redemptive plan for humanity.