Size your RichTextBox to 10 lines in height. ScrollToCaret will cause the RichTextBox to scroll to the bottom where you just added the line.
Code:
Public Class Form1
Dim i As Integer = 1
Private Sub uxAddLine_Click(ByVal sender As System.Object, ByVal e As System.EventArgs) _
Handles uxAddLine.Click
If Me.RichTextBox1.Text.Length = 0 Then
Me.RichTextBox1.AppendText("Line: " & i)
Else
Me.RichTextBox1.AppendText(Environment.NewLine & "Line: " & i)
End If
Me.RichTextBox1.ScrollToCaret()
i += 1
End Sub
End Class